
Cream 1 cup butter with 1 1/2 cups sugar, gradually adding the sugar and beating until light and fluffy. Add 2 eggs one at a time, beating after each addition. Stir in 1 1/2 teaspoons vanilla extract.
Combine 3 1/2 cups all-purpose flour, 1 teaspoon baking powder, and 1/2 teaspoon salt. Add flour mixture to creamed butter mixture, stirring well. Cover and refrigerate for at least 8 hours.
Work with one-fourth of dough at a time, storing remainder in refrigerator. Roll dough to 1/4- to 1/8-inch thickness on a lightly floured surface. Cut with assorted cookie cutters and place on ungreased cookie sheets.
Bake at 400°F (205°C) for 8 to 9 minutes. Cool cookies slightly on cookie sheets, then transfer to wire racks to cool completely.
Combine 1/2 cup softened butter and 1 (16-ounce) package powdered sugar, creaming until light and fluffy. Add 3 tablespoons half-and-half and 1 teaspoon vanilla extract. Beat until frosting reaches spreading consistency.
Color frosting with food coloring if desired. Spread frosting onto cooled cookies. Sprinkle with colored granulated sugar if desired.