It's reflecting sunlight right, so why does moonlight register as colder than shaded spots.
It's a real easy and fun experiment.
One guess is that the Moon is not a perfect reflector and the photon density per unit area coming off the Moon is a lot lower.
You could test this hypothesis by pointing a laser thermometer at a light bulb and a pile of dirt reflecting the light from the bulb. The Moon is basically made out of gray dirt.
It would be interesting to try the experiment on a moonless night to check the difference between open and shaded ground. If they are the same the shade doesn't likely matter (given a sufficient time from sundown).