Using an electric mixer with a whisk attachment, cream together the butter and sugar until fluffy. Add the egg, vanilla, and sour cream, and mix until thoroughly combined.
Divide the dough into 12 equal portions and place them on the prepared pan(s), leaving at least 2 inches between the cookies. Bake for 15 minutes, or until the edges are just beginning to brown.
Place the pan on a wire rack. While the cookies are still warm, either flip them over or use a flat-bottomed glass to flatten the tops. Allow the cookies to cool completely.