In a mixing bowl, combine sugar, butter, flour, and vanilla extract. Mixture will be crumbly. Press dough together with hands. Wrap tightly in plastic wrap and refrigerate for 1 hour.
Preheat oven to 350 degrees F. Roll out dough to 1/4” thick. Cut dough into rectangles. Place cutouts on a cookie sheet lined with a silicone mat or parchment paper.
Bake in preheated oven for 17 minutes, until edges are light golden brown. Let cool on a wire rack.
To assemble, spread a thin layer of frosting on the bottom of one cookie. Press another cookie on top of frosting. Repeat with remaining cookies. Store in an airtight container.