In a medium bowl, whisk together the flour, baking powder, and salt; set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugar on medium-high speed, until light and fluffy, about 2 minutes. Beat in egg and vanilla until incorporated. Reduce speed to low, add flour mixture, and mix until dough forms. Remove half of the dough from bowl and reserve. Add peppermint extract, candy, and food coloring to remaining dough and mix until combined.
Place the peppermint dough between 2 sheets of parchment paper and roll into 14 by 8-inch rectangle, chill until firm, about 15 minutes. Repeat with reserved plain dough. Once both layers of dough have been chilled, place the peppermint dough on top of the plain dough, pressing to adhere. With the long side facing you, roll the dough into a log. Wrap the dough in plastic wrap and refrigerate until firm, at least 2 hours.
Heat oven to 375°F. Line 2 baking sheets with parchment paper. Slice cookies into ¼-inch rounds and place on prepared baking sheets, spacing 1-inch apart. Bake until edges are just golden, 12 to 14 minutes, switching and rotating sheets halfway through baking. Let cool 10 minutes on sheet, then transfer to a wire rack and cool completely. Repeat with remaining dough.