Heat oven to 350° F.
Beat the butter and sugar in the bowl of an electric mixer on medium-high until smooth. Add the egg and beat until fluffy, about 2 minutes. Add the vanilla. Reduce mixer speed to low.
Combine the flour, baking soda, and salt in a bowl and add it slowly to the butter mixture, mixing until just incorporated (the dough will be stiff). Shape into a flat 1-inch-thick disk. Refrigerate, wrapped, for 30 minutes.
Roll out the dough on a floured surface to 1/4 inch thick. Cut into shapes and place on parchment-lined baking sheets. The scraps can be combined and rolled out again. Bake 10 to 12 minutes, until the edges are a light golden brown. Cool slightly on the baking sheets before transferring to cooling racks.