Line a rimmed baking sheet with parchment paper, and set aside. Sift the powdered sugar into a large mixing bowl, and set aside.

In the bowl of a stand mixer, beat the butter until light and fluffy, about 2 - 3 minutes. Slowly add the sifted powdered sugar on low speed, in batches, until fully incorporated.

Add the half and half, peppermint extract, vanilla extract, and salt to the mixture, beating on low speed until well-combined and the dough comes together in a ball. (If your dough is not coming together, add an additional teaspoon or so of half an half to bring it together.)

If coloring the mints, divide the dough into 3 equal portions. Add a few drops of food coloring to each portion and mix well until fully incorporated.

Wrap two of the portions in plastic wrap and set aside. This dough hardens quickly when exposed to air, and you want to keep it pliable.

Working on a nonstick surface, break off a piece of the unwrapped dough portion. Roll into a rope about ¾ inch in diameter. Slice into small pieces, ½ inch to ¾ inch long.

Transfer the cut pieces of dough to a baking sheet, and continue rolling and cutting the remaining dough.

Leave the butter mints out at room temperature to dry overnight. They should be dry and hard on the outside, but smooth and creamy on the inside.