Preheat the oven to 350°F. Line two baking sheets with parchment paper and set aside.
Mix the butter with a hand or stand mixer until creamy. Add the sugar and blend until light and fluffy. Stop the mixer and scrape down the sides of the bowl.
Add the egg, vanilla, and peppermint. Mix until combined.
Add the baking powder, salt, and baking soda. Mix until blended. Add the flour and mix on low speed until the dough comes together.
Divide the dough into three equal portions. Mix the coffee, pink dye, and cocoa powder - one into each of the three portions of dough.
Scoop about two teaspoons of each dough and roll into a ball. Smoosh them together and give the dough and twist. Roll into a smooth ball and toss in sugar.
Repeat with the remaining dough. Place on the lined baking sheets - at lease 2 inches apart.
Bake for 10 to 13 minutes. When done, the cookies should be crisp around the edges but still soft in the middle. Use a large round cookie cutter to scoot in any uneven edges.
Cool the cookies over a wire rack before serving and storing.