1. Make cookies: Mix flour, cocoa, baking soda and salt in a bowl. Using an electric mixer on medium speed, beat butter and sugar until light, about 3 minutes. Beat in egg and vanilla until smooth. Stir in flour mixture until just combined.
2. Divide dough in half. Shape each portion into a log about 9 inches long and 1 1/2 inches in diameter. Wrap in plastic; chill for at least 2 hours and up to 1 day (or freeze for up to 1 month).
3. Preheat oven to 350ºF. Place racks in upper and lower thirds of oven. Line 2 baking sheets with parchment. Slice dough into 1/4-inch-thick rounds (36 slices per log); place on baking sheets 2 inches apart. Bake until cookies are dry on top, about 10 minutes. Remove to a wire rack; cool. Repeat with remaining dough.
4. Make filling: Using an electric mixer on medium speed, beat butter, confectioners' sugar and salt until combined. Stir in Fluff and peppermint extract. Beat on high until light, about 3 minutes. Divide filling between 2 bowls, add red or green food coloring to each bowl; stir until evenly dyed. Sandwich 2 cookies with a teaspoon of filling. Repeat with remaining cookies and filling.