Cream butter, sugar, and vanilla in a mixer bowl on medium speed. Beat in flour and salt until just combined. Shape into a disc; wrap in plastic wrap; refrigerate until firm, at least 2 hours or up to overnight.
Line baking sheet with parchment paper. Roll out dough on a lightly floured surface to about 1/8 inch thick. Cut out with a 1 1/4-inch round cookie cutter; transfer to prepared pan. Freeze 30 minutes to harden.
Preheat oven to 325 degrees. Bake until cookies just begin to brown around the edges, 16 to 18 minutes. Let cool briefly on parchment, then transfer to a wire rack to cool completely.
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