In a medium bowl, whisk together the flour, baking powder, and salt. In the bowl of a stand mixer with paddle attachment, or separate large bowl with a hand mixer, cream together the butter and sugar until light and fluffy. Add the egg, vanilla extract, and peppermint extract, mixing until smooth. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until combined. The dough will be crumbly.
Shape the dough into balls, about 1 tablespoon each. Roll each ball in sprinkles, pressing lightly to adhere. If the sprinkles don’t stick well, mix a teaspoon of water into the dough to make it slightly tackier. Place the dough balls on a baking sheet or plate and refrigerate for at least 20 minutes.
Preheat the oven to 350°F. Line a baking sheet with parchment paper. Arrange the chilled dough balls on the prepared baking sheet, spacing them about 2 inches apart. Bake in the preheated oven for 12 minutes, until the edges are set but the centers still look slightly underdone.
Immediately press an unwrapped candy cane Kiss into the center of each cookie. Use a thin spatula to transfer the cookies to a plate, then place them in the freezer for 10 minutes. This step ensures the kisses hold their shape and don’t melt into the warm cookies.
Remove the cookies from the freezer and serve.