Melt butter; set aside to cool. Place cake mix, egg, cooled butter and vanilla in large mixing bowl. Mix by hand until well blended and dough is formed.
Form dough into a smooth 12-inch log on wax paper or parchment making sure it is even in thickness end to end. Wrap in wax paper; seal in plastic wrap. Refrigerate 1 hour or overnight.
Preheat oven to 350 degrees F. Lightly grease baking sheets. Unwrap dough log and set on wax paper. Using a sharp knife, cut into 48 slices about 1/4- inch thick. Place on cool baking sheets about 2 inches apart.
Bake, 1 sheet at a time, in center of oven for 10 to 12 minutes or until set. Cool 1 minute. Transfer to wire rack to cool completely.
Spread frosting on half the cookies. Top each frosted cookie with another cookie. Gently press together.