Preheat the oven to 300ºF. Line cookie sheets with parchment paper.
Make the cookie dough. Add all ingredients, in order they are listed, to food processor bowl. Pulse until a dough forms. The dough will be pretty crumbly, but this is fine. Turn out the mixture onto a piece of plastic wrap. Divide into two equal portions, form a ball, and smush down slightly into a disc. Wrap each tightly in plastic wrap and chill for about 10 minutes.
Remove dough from the fridge, unwrap, and roll out to ⅛" thickness in between two sheets of parchment paper.
Cut out cookies with round cookie cutter, about 2 ½" wide or so. Place the cookies about 1" apart on the parchment-paper-lined cookie sheet. Return to fridge or freezer for about 5 to 10 minutes, or while you continue to cut out the rest of the cookies.
After chilling, remove cookie sheet and place in preheated oven. Bake about 15 minutes, or until firm to the touch. Note: if the cookies ever begin to smell nutty, they're going to burn! Remove them from the oven.
Carefully transfer to wire cooling rack and cool completely. Meanwhile, make peppermint filling.