Stir together flour, baking powder, and salt in a large bowl.

With an electric mixer on low speed, beat butter and sugar in a medium bowl until well blended. Add egg, egg yolk, vanilla, and peppermint extract; beat until smooth. Add the flour mixture and beat just until blended. Transfer half of the dough to a work surface. Beat in the food coloring into the remaining dough until evenly colored.

Pat the colored dough into a rectangle on a 12-inch long piece of plastic wrap. Cover with another piece of plastic wrap. Roll out the dough to form a 6 x 12‑inch rectangle. Repeat with the remaining dough. Refrigerate the dough for 10 minutes. Remove the top sheets of plastic wrap from each rectangle of dough. Flip the colored dough on top of the uncolored dough; remove the plastic wrap. Using the bottom sheet of plastic wrap to help you roll, and starting from the long side, roll up the dough, jelly-roll style, to form a tight log. Wrap in the plastic wrap and refrigerate until firm, at least 2 hours.

Preheat oven to 350°F. Line two large baking sheets with parchment paper or foil.

Cut the log into 1/4-inch slices making a total of 24 slices. Place on the baking sheets about 1-inch apart. Insert a lollipop stick (about 3/4-inch deep) into the bottom of each cookie. Bake until the cookies are set, about 12 minutes, rotating the baking sheets halfway through the baking. With a spatula, transfer the cookies to racks to cool completely.