Prepare the chestnuts by cutting a small x on the top of each one with a very sharp knife. It's better to go all the way through the shell into the flesh of the chestnut than to not cut them deep enough. Arrange on a baking sheet, sliced sides up, and bake for 20-30 minutes, until they turn darker brown and the sliced edges peel back to reveal the chestnut inside. Allow to cool and then peel them. It's ok if they break while peeling.
Once the chestnuts are fully cooled, chop them coarsely. Measure out one cup of the chestnuts and place in the bowl of a food processor (see note). Pulse until very well chopped. Add the softened butter, and pulse to combine. Add 1/2 cup of powdered sugar, vanilla extract, 1/4 teaspoon of cinnamon, the nutmeg, salt, and flour. Pulse to form an even dough. Turn the dough out onto a clean surface and divide in two. Wrap each half in plastic wrap, and refrigerate until firm, about 1 hour.
Preheat oven to 350˚.
Combine remaiming 1 1/2 cups of powdered sugar with 1/4 teaspoon of cinnamon in a small bowl. Working with 1 batch of chilled dough at a time, roll the dough into 2-teaspoon-sized balls. Arrange on a baking sheet lined with parchment. Bake 14-17 minutes, until just pale golden on top and golden-brown on the bottom. Cool 4-6 minutes on the baking sheet, and then toss the cookies in the powdered sugar mixture. Place the sugar-coated cookies on a wire rack to cook completely. Before serving, sift any remaining powdered sugar mixture over the cookies.
If you don't have a food processor: Chop the chestnuts as fine as you can. Whip the butter and 1/2 cup powdered sugar in a large bowl. Add vanilla, 1/4 tsp. cinnamon, nutmeg, salt, flour, and chopped chestnuts, and blend well. Continue with recipe.