1. Make the cakes: Preheat oven to 350 degrees. Butter two 12-by-17-inch rimmed baking sheets. Line with parchment. Butter parchment, and dust with flour, tapping out excess. Sift flour, baking powder, and 1 1/2 teaspoons salt into a bowl.

2. Combine milk and vanilla. Beat butter with a mixer on medium speed until very smooth. With machine running, gradually add sugar, and beat until pale and fluffy, about 3 minutes.

3. Reduce speed to low. Add flour mixture in 3 additions, alternating with milk mixture, beginning and ending with flour. Beat until just combined; do not overmix.

4. Beat egg whites in a clean mixer bowl on medium speed until stiff peaks form, about 3 minutes. Gently fold egg whites into batter in 3 additions.

5. Transfer batter to baking sheets, smoothing tops with an offset spatula. Bake until tops spring back when pressed with a finger and a toothpick inserted into the centers comes out clean, about 30 minutes. Let cool in sheets on wire racks for 15 minutes. Loosen edges of cakes with a small metal spatula or paring knife; invert onto racks. Discard parchment. To prevent splitting, reinvert cakes, top side up. Let cool completely.

6. Decorate the cakes: Place "20" stencil (or another number) onto 1 cake, setting numbers side by side. Secure with toothpicks. Cut around each stencil using a paring knife. Remove scraps. Repeat with remaining cake.

7. Spread 2 cups buttercream onto the "2," and spread 2 cups buttercream onto the "0." Top each number with remaining matching cake number, and frost tops and sides of each with remaining buttercream. Cover each cake with rows of candies.