The wikipedia article on microwave ovens says
Another misconception is that microwave ovens cook food "from the inside out", meaning from the center of the entire mass of food outwards.
It further says that
with uniformly structured or reasonably homogenous food item, microwaves are absorbed in the outer layers of the item at a similar level to that of the inner layers.
However, on more than one occasion I've microwaved a stick of butter, and the inside melts first, then the outside caves in releasing a flood of butter. (It may be relevant that my microwave turntable does not turn – but since I've done it more than once, I would not expect it to be a fluke of placement in the standing wave. And, the resulting butter-softness seemed very strongly correlated with depth, more than I'd expect from accident.) That sure seems consistent with the food absorbing more energy on the inside than on the outside. Given that this takes place over 30 seconds or so, I'd not expect much heat exchange to occur with the butter's environment (nor inside the butter itself), so that would forestall explanations of "the air cools off the outer layer of butter", unless I'm seriously underestimating the ability of air to cool off warm butter. So what's going on?
Best Answer
A microwave oven's microwaves have a half-wavelength of about 6 centimeters. While there are microwaves bouncing around in all directions and with all manner of offset origins inside the mirrored (for microwaves) interior of the oven, there's still a significant contribution to the energy deposited in the target that's from the primary beam.
A stick of butter is around 12 centimeters long. Therefore, it's highly likely that a randomly positioned stick of butter has exactly one maximum amplitude point located somewhere between the end points. This point will receive the most energy and melt first. Because butter has a very low melting point, it only needs to receive a little bit more energy to melt first, and because it's not in a rigid container, the melted part can puddle down and expose the part that was previously shielded from the radiation by the outer layer to the same maximum amplitude point.
This does not in any way contradict the claim that microwaves cook from the outside in. They simply cook from the outside-in slightly more where the primary wave direct from the magnetron is bigger. A working turntable helps to reduce this effect to make for more even heating across the whole target.
Air cooling is almost entirely irrelevant.
As an aside - you can do a crude measurement of the speed of light if you line up two sticks of butter in a row, measure the distance $\lambda / 2$ between the places where they first start to melt, and multiply by twice the frequency $\nu$ listed by the manufacturer of the oven. $c = \lambda \nu$