Put a piece of parchment or Silpat on the counter. Invert the cake onto it and peel off the parchment or Silpat from the bottom of the cake. Use the cake ring to stamp out 2 circles from the cake. These are your top 2 cake layers. The remaining cake "scrap" will come together to make the bottom layer of the cake.
Layer 1, The Bottom: Clean the cake ring and place it in the center of a sheet pan lined with clean parchment or a Silpat. Use 1 strip of acetate to line the inside of the cake ring.
Put the cake scraps in the ring and use the back of your hand to tamp the scraps together into a flat even layer.
In a small bowl, whisk together the milk and vanilla extract for the birthday cake soak. Using a pastry brush, brush half of the soak over the top of the cake.
Using the back of a spoon, spread one-fifth of the frosting in an even layer over the cake.
Sprinkle one-third of the birthday cake crumbs evenly over the top of the frosting. use the back of your hand to press them gently into the frosting.
Use the back of a spoon to spread a second fifth of the frosting as evenly as possible over the crumbs.
Layer 2, The Middle: With your index finger, gently tuck the second strip of acetate between the cake ring and the top 1/4 inch of the first strip of acetate, so that you have a clear ring of acetate 5 to 6 inches tall. Set one of the cake rounds (the more imperfect of the two) on top of the frosting, brush with the remaining soak, and repeat the process for Layer 1 (a fifth of frosting, a third of crumbs, and another fifth of frosting).
Layer 3, The Top: Nestle the remaining cake round into the frosting. Cover the top of the cake with the last fifth of the frosting. Garnish the frosting with the remaining birthday crumbs.
Transfer the sheet pan to the freezer and freeze for a minimum of 12 hours to set the cake and filling. (The cake will keep in the freezer for up to 2 weeks.)
At least 3 hours before you are ready to serve the cake, pull the sheet pan out of the freezer and, using your fingers and thumbs, pop the cake out of the cake ring. Gently peel off the acetate and transfer the cake to a platter or cake stand. Let it defrost in the refrigerator for a minimum of 3 hours before slicing and serving. (You can store the cake, wrapped well in plastic wrap, in the refrigerator for up to 5 days.)