Whisk together the flour, baking powder, and salt in a medium bowl to combine.
Beat the butter, shortening, and sugar in the bowl of an electric mixer on medium speed for 2 to 3 minutes, until light and fluffy. Add the egg and vanilla extract and beat until well combined. With the mixer on low speed, add the flour mixture in two additions, blending until just incorporated. Cover the dough with plastic wrap and chill for about 2 hours, until firm.
Preheat the oven to 325°F and line a few baking sheets with parchment paper. Transfer the dough to a well-floured work surface and roll out to ¼-inch thickness. Cut out shapes of dough and arrange on the prepared baking sheets, spacing the cookies 1-inch apart. Bake for about 14 minutes, until the cookies are just firm to the touch. Remove from the oven and let cool for 5 minutes. Transfer the cookies to a wire rack and let cool completely.
Beat the butter, powdered sugar, milk, and vanilla extract in the bowl of an electric mixer on medium speed for 1 to 2 minutes, until smooth and creamy. Add some food coloring, a drop at a time, and mix until the frosting reaches the color you want. Spread about 1 teaspoon of frosting over each cookie and decorate with sprinkles.