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To make the cookies: In the bowl of a stand mixer fitted with the flat beater attachment or working in a large bowl with an electric hand mixer, 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.
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Divide the dough in half, shape each half into a flattened disk, and wrap in plastic. Refrigerate for 2 hours, or overnight.
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When you're ready to bake, preheat the oven to 350°F. 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.
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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.
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Bake the butter cookies for 12 to 14 minutes, until they're set and barely browned around the edges.
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Remove the butter cookies from the oven and allow them to 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.
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Repeat with the remaining piece of dough, rolling, cutting, and baking the cookies. When the butter cookies are completely cool, ice and decorate as desired.
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To make the icing: In a medium bowl, combine the sugar, 2 1/2 tablespoons (35g) of the milk, and corn syrup to make a soft, spreadable icing. Add more milk, a teaspoon at a time, if the frosting is stiff. Tint the icing with food color as desired.
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Test the consistency of the icing by using a knife, spoon, or offset spatula to spread it generously over the surface of one cookie. The frosting will be thick but should smooth out after a minute or so. Add additional milk or confectioner's sugar as necessary until it's the right consistency for your decorating needs.
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Decorate the cookies a few at a time, sprinkling coarse sparkling sugar over the top of the icing while it's still wet so the sugar adheres. Set the cookies aside until the icing has set and feels dry to the touch.
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Storage information: Store leftover butter cookies at room temperature in an airtight container for several days; freeze for longer storage.
