1) To make the cookies: Combine the sugar, butter, egg yolk, salt, and flavor, beating until smooth. Add the flour, mixing until smooth. The mixture will seem dry at first, but will suddenly come together. If it doesn't, dribble in a tablespoon of water.
2) Divide the dough in half, shape each half into a flattened disk, and wrap in plastic. Refrigerate for 2 hours, or overnight. When you're ready to bake, remove the dough from the refrigerator, and let it soften for about 20 to 30 minutes, until it feels soft enough to roll. It should still feel cold, but shouldn't feel rock-hard.
3) Sprinkle your rolling surface with flour, and flour your rolling pin. Working with one piece of dough at a time, roll it 1/8" to 3/16" thick. Use a cookie cutter to cut shapes. Re-roll and cut the dough scraps. Place the cookies on ungreased or parchment-lined baking sheets. They can be close together; they'll barely spread.
4) Bake the cookies in a preheated 350°F oven for 12 to 14 minutes, until they're set and barely browned around the edges.
5) Remove the cookies from the oven, and cool right on the pan. If you've used parchment, you can lift cookies and parchment off the pan, so you can continue to use the pan as the cookies cool.
6) Repeat with the remaining piece of dough, rolling, cutting, and baking cookies. When cookies are completely cool, ice and decorate.
7) Icing: Combine the sugar, milk, and corn syrup to make a soft, spreadable icing, adding more milk if necessary. Tint the icing with food color as desired.
8) Spread icing on the cookies, using a knife, a spoon, or your finger to spread it all the way to the edges. Sprinkle with colored sugar or other sugar decorations, as desired. Allow the icing to harden before storing the cookies.
Yield: about 5 dozen 2" cookies.