I'd like to expand my earlier comment into a little essay on the severe practical difficulties in performing the suggested experiment.
I'm going to start my asserting that we don't care if the experiment is a "two-slit" per se. It is sufficient that it is a diffractive scattering experiment of some kind.
However, we do care about having
spacial resolution good enough to distinguish which scattering site (or slit) was the one on the path of the alleged particle
the ability to run the experiment at low rate so that we can exclude multi-projectile or beam/beam interaction as the source of any interference that we observe. (Though it's going to turn out that we never even get far enough for this to matter...)
Now let's get down to designing the beast.
To start with we should note to any casual readers that the diagrams you see in pop-sci treatment are not even remotely to scale: typical classroom demonstration kit for use with lasers has the slits set less than $1\,\mathrm{mm}$ apart and uses projection distances of several meters or more to get fringes that are separated by a few centimeters. Or then use much closer set slits to get large angles.
The angular separation between maxima is on order of
$$ \Delta \theta = \frac{\lambda}{d} \,,$$
where $\lambda$ is the relevant wavelength and $d$ is the scattering site (or slit) separation. Allowing that the distance from the scattering surface to the projection surface is $\ell$, the spacial separation is (in the small angle approximation)
$$ \Delta x = l \Delta \theta = \frac{\ell}{d} \lambda \,.$$
Anna has suggested doing the experiment with electrons, which means that we're interested in the de Broglie wavelength usually given by $\lambda = \hbar/p$, and measuring their position en route with a tracking detector of some kind.
The tracking detector's spacial resolution is going to be the big barrier here.
Let's start by considering a Liquid Argon TPC because it is a hot technology just now. Spacial resolution down to about $1 \,\mathrm{mm}$ should be achievable without any breakthrough in technology (typical devices have $3$-$5\,\mathrm{mm}$ resolution). That sets our value for $d$.
Now, to observe a interferences pattern, we need a detector resolution at least four times finer than the spacial resolution.
Assume for the sake of argument that I use a detector with a $20 \,\mathrm{\mu{}m}$ spacial resolution. Maybe a MCP or a silicon tracker. That sets $\Delta x = 4(20 \,\mathrm{\mu{}m})$.
I also assume that I need $\ell$ to be at least $2d$ to be able to track the particle between the scattering and projection planes. Probably an under-estimate, so be it. Now I can compute the properties of the necessary electron source
$$\begin{align*}
p &= \frac{\hbar}{\lambda} \\
&= \frac{\hbar\ell}{d \, \Delta x} \tag{1}\\
&= 2\frac{\hbar}{\Delta x}\\
&= \frac{7 \times 10^{-22} \,\mathrm{MeV \, s}}{40 \times 10^{-6} \,\mathrm{m}}\\
&= \frac{7 \times 10^{-22} \,\mathrm{MeV}}{7 \times 10^{-12} c} \\
&= 10^{-10} \,\mathrm{MeV/c}\\
&= 10^{-4} \,\mathrm{eV/c} \,,
\end{align*}$$
which is safely non-relativistic, so we have a beam energy of $5 \times 10^{-9}\,\mathrm{eV^2}/(m_e c^2)$, and the tracking medium will completely mess up the experiment.
By choosing a $20\,\mathrm{m}$ flight path between scattering and detection and
getting down to, say, the $10\,\mathrm{\mu{}m}$ scale for $d$ we can get beam momenta up to $10^3\,\mathrm{eV}$ which at lest gives us beam energies about $1\,\mathrm{eV}$. But how are you going to track a $1\,\mathrm{eV}$ electron without scattering it?
I'm sure you can get better spacial resolution in silicon, but I don't think you can get the beam energy up high enough to pass a great enough distance through the tracking medium to actually make the measurement.
The fundamental problem here is the tension between the desire to track the electron on it's route which forces you to use nearly human scales for parts of the detector and the presence of that pesky $\hbar$ in the numerator of equation (1) which is driving the necessary beam momentum down.
The usual method of getting diffractive effects is just to make $d$ small and $\ell$ large enough to compensate for the $\hbar$ but our desire to track the particles works against us there by putting a floor on our attemtps to shrink $d$ and by because longer flight paths mean more sensitivity to scattering by the tracking medium.
Best Answer
It is not possible to be completely sure there is only one photon in the apparatus and in the quantum sense this is even an ambivalent question to ask. But to the question "Is there a source of photons for which we almost always detect one outcoming photon in a given time interval?" the answer is yes, single photon sources. As far as I know, it is however always a continuous source of photons, that is, it keeps spitting "antibunched" photons out continuously and just a single photon cannot be produced.
As far as I know, experiments with single photon sources have however not found any deviation from quantum mechanical predictions and the "single photons" behave completely as quantum particles. Historical electron double slit experiments were usually done in vacuum.
I don't know whether a "moving slit" experiment has been done, but there are some issues with it such as collecting a large number of dots forming the interference pattern. Obviously, you would like to get the full pattern of a photon which came out at $t_0$ and the double slit was moved from $x(t_0)$ to $x(t_0)+v\Delta t$. However, in a naive setting you will get a superposition of patterns from double slits at different position as the source keeps spitting out photons at different times when the slit is at different initial positions.
The time $t_0$ of the "photon spit" is generally also not settable accurately due to quantum principles. So in any case, I believe the resulting image would basically be blurred and inconclusive which is the reason the experiment has probably never been done.
As to a formal treatment, we could idealise the situation by taking the free particle with boundary conditions of the classical double-slit experiment and making the boundary conditions of the double slit time dependent and solve the time-dependent wave equation (a massless Klein-Gordon equation would be okay). However, as typical corrections of the perturbed solution would be $v/c$, where $v$ is the velocity of the double slit, a quasi-static treatment would most probably be sufficient. The probability density of finding a photon on the display would then just be the interference pattern of the non-moving double slit experiment translated in time according to the translation of the double slit.
Detecting the time of the incidence of the display could in principle resolve the issue of blurriness and a repeated super-sensitive experiment might reconstruct the time-frozen interference pattern and investigate any deviations from the predictions of a less idealized time-dependent wave equation (accounting e.g. for the interaction with the slit wall). Nevertheless, keep in mind that we do not know any kind of classical trajectory of a photon and thus do not know the time it took to go through the apparatus, and we also do not know at which exact time it left the source, so any formulation of alternative predictions would be very difficult.
EDIT: There is indeed a theory defining sharp states of quantum particles, the De Broglie-Bohm theory and in the non-static case, the formulation using the quantum potential would perhaps be useful.
However, these so called "hidden variable theories" also just speak in terms of probability distributions since any attempt to measure the particle state seriously disturbs the "piloting wave".
Just imagine how to actually measure the photon initial time and position, you cannot "see" it, you have to "bump" into it. But with what we can "bump" into it? Say with another photon, whose initial position and fly-out time we once again do not know! This "measurement" process can only converge to a limited certainty expressed by the Heisenberg uncertainty principle. This follows from the fact that there is a lower limit on the size of the particle/wavepacket you can use for measurement.
So in our case, if we tried to measure the initial momentum and position of the photon at a given time, our "probing" particle just could not retrieve everything without disturbing the certainty in other variables. This is why in the end De-Broglie Bohm theory reproduces all the results of quantum mechanics almost to the dot.
The term "frozen interference patterns" refers only to the fact that our wave-function will be time-dependent:
$\psi(x,t)$
and so the probability density of finding a particle on a given position on the display $x_d$:
$\rho(x_d,t) = |\psi(x_d,t)|^2$.
(In the model I have proposed, it will just be an interference patter moving along the display). However, in one run of the experiment, we will only get one dot for every $t_1,t_2,t_3,...$ on the display. We would however like to recover $\rho(x_d,t_1),\rho(x_d,t_2)...$ (the "time-frozen" interference patterns) which is possible only with very carefully rerunning the experiment again and again and getting more and more dots at various times. We can then get e.g. the "time-frozen" $\rho(x_d,t_1)$ by plotting only detections which happened at $t_1$.