Your question is an interesting one, but I think the reason you haven't received an answer to it yet is that it's ill-constrained. That's not a criticism but a technical term. Basically it means that the problem isn't well-defined enough for there to be a clear, distinct answer. (This is what the second part of Rahul Narain's comment is getting at.) To borrow from the Math Overflow FAQ: "The site works best for well-defined questions: math questions that actually have a specific answer."
Instead, your question would be great in my mathematical modeling course. I would pose it and then ask the students what assumptions would need to be made in order to get a well-defined mathematical model, or what aspects of the problem would be parameters to the model that we could vary and then see how the solution changes (such as those Rahul Narain mentions), or what existing mathematical tools we could use to help understand traffic behavior (such as queuing theory, as Yuval Filmus mentions). An answer to the question would then require some student taking this on as an extended assignment. He or she would have to make and justify assumptions, create a model (or two or three, which might require some programming), testing it (or them), refine the model(s), vary parameters, and then interpret the output from the model(s). There probably wouldn't be a single answer but a range of answers along the lines of "Based on these assumptions our model says this."
(I've actually just described the process of mathematical modeling.)
Another aspect of your question is that traffic problems are hard. They frequently show up as problems in the Mathematical Contest in Modeling (see, for example, the 2009 and 2005 contests).
Traffic problems are an active area of research, too, because there's a lot we don't understand about the way traffic behaves. There's not even a consensus about the best way to go about modeling traffic. Fluid flow models seem to be the most popular, but some people use queuing, and I've also seen discrete dynamical system and even cellular automata models.
If you're interested in learning more about research on traffic, you could also check out this article ("Traffic jam mystery solved by mathematicians") or the book Mathematical Theories of Traffic Flow.
This question could be open to interpretation but my reading of it is that when the pedestrian arrives at the road he can immediately see the road from where he stands to $k$ time units along the road, so that he can identify a coming time gap of $k$ units. So, if this section of road is empty at that time, he crosses immediately so that $T=0$ and the total time taken for the crossing is $T+k=k$.
Let $X$ be the time of the first car arrival and condition on the event $X\lt k$, meaning that the next car lies in that $k$-length section of road:
\begin{align}
E(T) &= P(X\lt k)E(T\mid X\lt k) + P(x\geq k)E(T\mid X\geq k) \\
& \\
&= P(X\lt k)\left[E(T) + E(X\mid X\lt k)\right] + P(x\geq k)\cdot 0 \\
& \qquad\qquad\text{since, if $X\lt k$, we count $X$ and then re-start the wait} \\
& \qquad\qquad\text{and if $X\geq k$, there is no waiting required} \\
& \\
\therefore\quad E(T) &= \dfrac{1}{1-P(X\lt k)} P(X\lt k) E(X\mid X\lt k) \\
& \\
&= \dfrac{1}{e^{-\lambda k}} \int_{x=0}^{k} \lambda x e^{-\lambda x}\;dx \\
& \\
&= e^{\lambda k} \left[ e^{-\lambda x}\left( -x-1/\lambda \right)\right]_{x=0}^{k} \\
& \\
&= \dfrac{1}{\lambda}\left( e^{\lambda k} - 1 \right) - k.
\end{align}
Therefore, the mean time to complete the crossing is:
$$E(T) + k = \dfrac{1}{\lambda}\left( e^{\lambda k} - 1 \right).$$
Best Answer
The chicken waits between the passages of car $i-1$ and car $i$ without crossing the road if and only if no gap between car $n-1$ and car $n$ is greater than $k$, for any $n\leqslant i$. Thus, $$ T=\sum_{i=1}^{+\infty}D_i\cdot[D_1\leqslant k,\ldots,D_i\leqslant k], $$ where $(D_i)_{i\geqslant 1}$ is i.i.d. with exponential distribution of parameter $\lambda$. By independence, $$ \mathrm E(D_i;D_1\leqslant k,\ldots,D_i\leqslant k)=a^{i-1}b,\quad a=\mathrm P(D\leqslant k),\quad b=\mathrm E(D;D\leqslant k), $$ hence $\mathrm E(T)=b/(1-a)$. One knows that $1-a=\mathrm P(D\gt k)=\mathrm e^{-\lambda k}$, and $$ b=\int_0^kx\cdot\lambda\mathrm e^{-\lambda x}\cdot\mathrm dx=\frac{1-(1+\lambda k)\mathrm e^{-\lambda k}}\lambda. $$ Finally, $$ \mathrm E(T)=\frac{\mathrm e^{\lambda k}-1-\lambda k}\lambda. $$