1/2 cup (1 stick) unsalted butter, softened
3/4 cup sugar
1 teaspoon vanilla
2 cups all-purpose flour
1 teaspoon baking powder
1/2 teaspoon salt
Cream the butter and sugar. Add the egg and beat until fluffy. Blend in the vanilla. Sift in the flour, baking powder, and salt, mixing until just combined. Wrap the dough in plastic wrap and chill for 1 hour.
Preheat the oven to 325 degrees F.
Roll out the dough to 1/4 to 1/2-inch thick. Place the cookies on a greased baking sheet and bake for about 10 minutes, until just turning golden. Do not allow cookies to brown. (they will probably need a little longer as they will be MUCH thicker than normal).
Cool completely if decorating.
(for the record, I usually cook mine a little hotter, 350, but it is probably differences in the oven. I also add a little mint when I mix the dough)