In a bowl, with an electric mixer on medium speed, beat butter and sugar until smooth. Beat in egg yolks and vanilla. Stir or beat in flour until well blended.
Divide dough in half. Flatten each portion into a 1-inch-thick disk. On a lightly floured board, with a floured rolling pin, roll dough, a portion at a time, to 1/8 inch thick.
With floured cookie cutters, cut dough into shapes (if dough is too soft to handle, freeze briefly until firm).
With a wide spatula or your fingers, transfer cookies to buttered or cooking parchment-lined 12- by 15-inch baking sheets, spacing them about 1 inch apart. Gather scraps, pat into a ball, and repeat rolling and cutting.
Bake cookies in a 300° oven until golden, about 15 minutes; switch pan positions halfway through baking.
With a wide spatula, transfer cookies to racks to cool. If hot cookies start to break, slide a thin spatula under them to release; let stand on sheets to firm up, about 5 minutes, then transfer to racks to cool completely.
Powdered sugar icing. Controlling consistency of icing is simple. If it's too thick, add a few drops of water; if it's too thin, gradually add more powdered sugar. To tint, add food coloring a few drops at a time.
For spreading: In a bowl, stir 3 cups powdered sugar, 1 teaspoon vanilla, and 1/4 cup water until smooth. Tint with food coloring as desired. Spread icing onto cool cookies with a metal spatula. Let stand until icing is firm, about 1 1/2 hours. Makes about 1 cup, enough to coat 1 recipe of Favorite Cutout Cookies.
For piping: In a bowl, stir 2 cups powdered sugar, 1/2 teaspoon vanilla, and 1 1/2 tablespoons water until smooth. Tint with food coloring as desired. Spoon into a pastry bag fitted with a 1/8-inch tip or a heavy 1-quart zip-lock plastic bag. Seal plastic bag and squeeze icing into a corner; cut a tiny opening at corner of bag to squeeze icing out. Let cookies stand until icing is firm, about 5 minutes. Makes about 3/4 cup, enough for 1 recipe Favorite Cutout Cookies.