In a medium bowl, whisk together flour, baking powder and salt; set aside. In a large bowl, beat butter, sugar and vanilla with an electric mixer at medium speed until light and fluffy. Beat in egg until well blended. Add flour mixture and beat on low speed until combined. Divide dough in half and shape into two discs. Wrap in waxed paper and chill until firm, at least 1 hour or overnight.
Preheat oven to 350°F. Working with one disc of dough at a time, roll out dough on a floured surface until (1/4-inch) thick and cut out cookies with floured cookie cutters. Transfer cookies to parchment paper-lined baking sheets, spacing them 1 inch apart.
Bake until just golden around the edges, 10 to 12 minutes. Transfer cookies to a wire rack to cool and then decorate as desired. Store in an airtight container at room temperature.