Beat butter and vanilla at medium speed with an electric mixer until creamy. Whisk together powdered sugar, flour, and baking powder. Gradually add sugar mixture to butter mixture, beating at low speed until blended. Flatten dough into a disk, and wrap in plastic wrap. Chill 1 hour to 3 days.
Preheat oven to 350°. Place dough on a lightly floured surface; roll to 1/4-inch thickness. Cut with desired cookie cutters. Place 8 cookies 1/2 inch apart on a parchment-lined baking sheet.
Bake at 350° for 12 to 13 minutes or until edges are lightly browned. Cool on baking sheets 5 minutes. Transfer to wire racks; cool completely (about 20 minutes).
Follow the wet-on-wet icing technique to decorate cookies (see Elegantly Ice with Ease, below). Let dry 1 hour.
Elegantly Ice with Ease
Using royal icing (such as Thin Royal Icing) with a wet consistency, pipe a thin border around the edge of the cookie. Fill in the center of the cookie with an even layer of icing.
To smooth the surface, swirl a wooden pick in a circular motion around the entire pool of icing, starting at the edges and moving toward the center.
Immediately pipe another color of icing, the same consistency as the first, onto the base layer of icing. Manipulate the design gently with a wooden pick.