Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper and set them aside.
In a large bowl, beat the softened butter until smooth. Add the granulated sugar and beat until light and fluffy. Scrape down the sides of the bowl with a spatula.
Add the egg, vanilla extract, and peppermint extract to the butter mixture. Beat until well combined.
In a separate bowl, whisk together the all-purpose flour, baking powder, salt, and baking soda. Add the dry ingredients to the wet ingredients and mix on low speed until a soft dough forms.
Divide the dough evenly into three portions. Knead the instant coffee into the first portion, the cocoa powder into the second portion, and the pink food coloring into the third portion.
Scoop about two teaspoons of each dough color. Press the three colors together, twist gently, and roll into a single smooth ball. Roll the combined dough ball in extra granulated sugar.
Place the cookies on the prepared baking sheets, leaving at least 2 inches of space between each cookie.
Bake for 10 to 13 minutes, or until the edges are firm but the centers are still soft. For food safety, baked goods with eggs should reach an internal temperature of 160°F (71°C).
Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.