Place a silicone baking mat or piece of parchment on a counter. Invert cake onto mat, then peel off parchment. Use cake ring to punch out two 6" rounds from cake (or, using a springform pan as a guide, cut out 2 rounds using a paring knife). These are your top 2 cake layers (the remaining cake scraps will form the bottom layer of the cake).
Line a sheet pan with a clean piece of parchment. Clean cake ring and place it in center of the pan. Use 1 acetate strip to line inside of cake ring. Place cake scraps inside ring and use the back of your hand to press scraps together into a flat, even layer (you never see this layer, so it’s okay that it’s messy—but since it’s the base of the cake, it needs to be flat).
Combine milk and remaining 1 tsp. vanilla in a small bowl. Dunk a pastry brush in milk mixture and use half of it to generously moisten the base layer.
Use the back of a spoon to spread about 3 Tbsp. frosting evenly over cake. Sprinkle ⅔ cup birthday crumbs evenly over frosting. Use the back of your hand to press them in place. Use the back of a spoon to spread another 3 Tbsp. frosting as evenly as possible over crumbs.
With your index finger, gently tuck second acetate strip between cake ring and the top ¼" of the first acetate strip, so that you have a clear ring of acetate 5–6" tall—high enough to support the height of the finished cake. Top with a cake round (if 1 of your 2 cake rounds is less pretty than the other, use it for the middle layer and save the most perfect one for the top). Brush layer with remaining milk mixture. Repeat frosting-crumb layering process.
Nestle remaining cake round into frosting. Cover top of cake with remaining frosting. Use an offset spatula to form decorative swirls, or do as they do at Milk Bar and shape it into a perfectly flat top. Top with remaining birthday crumbs.
Transfer cake to freezer and freeze at least 3 hours to set cake and filling.
At least 3 hours before serving the cake, pull sheet pan out of freezer and, using your fingers and thumbs, pop cake out of cake ring. Gently peel off acetate and transfer cake to a platter or cake stand. Defrost in fridge at least 3 hours.