An adaptation of the famous Milk Bar birthday cake: melt-in-your-mouth layers of Funfetti cake with birthday cake soak, topped with vanilla cream cheese frosting and birthday cake crumbs.
Preheat oven to 300 degrees Fahrenheit and line a rimmed baking sheet with parchment paper.
Using a stand mixer fitted with the paddle attachment, mix together the sugars, flour, baking powder, salt, and sprinkles on low speed until combined. Add the oil and vanilla and mix until it begins to come together and small clusters form.
Use your hands to make more clusters as needed, and transfer crumbs to the baking sheet. Bake for 20 minutes, stirring halfway through. Remove from the oven and allow to cool completely. If making the crumbs ahead of time, they can be stored for a few days in an airtight container at room temperature.
Preheat the oven to 350 degrees Fahrenheit and line the bottom of a 10x15 inch, rimmed quarter sheet pan with parchment paper. Spray the pan generously with cooking spray.
In a medium-size bowl, whisk together the flour, baking powder, salt, and 1/4 cup of the sprinkles. Set aside. In a measuring cup, combine the buttermilk, oil, and vanilla extract. Set aside.
Using an electric mixer fitted with the whisk attachment, cream together the butter and both sugars on medium-high speed for about 3 minutes. Add the eggs and beat for another 2-3 minutes.
Reduce to low speed and slowly pour in the buttermilk mixture. Increase speed to medium-high and beat for about 4 more minutes until mixture is smooth, shiny, and increased in size.
Reduce speed to low again and add the flour mixture gradually, mixing until just combined. Pour batter into the prepared baking pan and smooth evenly with a spatula. Distribute remaining 2 tablespoons of sprinkles evenly on top of the batter.
Bake at 350 degrees for 28-32 minutes, or until cake is golden brown and toothpick inserted into the center of the cake comes out clean. Remove from oven and place the pan on top of a wire rack to cool completely.
Using a stand mixer fitted with the paddle attachment, beat together the butter and cream cheese on medium speed until smooth. Gradually add in the powdered sugar, beating on low to combine.
Add the vanilla extract, milk, pinch of salt, and squeeze of lemon juice, then mix to combine. Beat on medium speed until smooth.
Whisk together the milk and vanilla extract in a small bowl or measuring cup.
Invert the cake pan on a cutting board and remove the parchment paper from the bottom. Using a six-inch cake ring, stamp out two circles from the cake. The scrap will form the bottom layer of the cake (I cut out as close to a full circle as possible, then a small piece for the remaining edge of the circle).
Tape a cake board onto a sheet pan and place the cake ring around it. Line the inside edge of the cake ring with a strip of acetate.
Press the scraps together inside the ring to form the bottom layer. The top of the cake (part facing up when baking) should be facing *down*. Using a pastry brush, soak the bottom layer with half the birthday cake soak.
Spread 1/5th of the frosting on top with the back of a spoon, then top with 1/3rd of the birthday cake crumbs, pressing them down gently. Spread another fifth of the frosting on top (be patient here - I find this to be the most frustrating part. It helps if you put a big dollop of frosting in, then spread gently from the inside toward the outer edges. It doesn't need to be perfect).
Top with the second layer of cake (face down again), brush with remaining milk soak, then repeat step 4. If needed, add another strip of acetate (tucked between the cake ring and the very top of the first strip to increase the height to 5 or 6 inches).
Stack the final layer of cake (top side facing up this time!) and cover with remaining frosting. Arrange the remaining birthday cake crumbs on top.
Transfer the cake to the freezer and freeze overnight (at least 8-12 hours) to set the layers. About 3 hours before you want to serve the cake, remove from the cake ring and peel off the acetate. Transfer to the fridge to let it defrost.
* To make your own cake flour: sift together 1 3/4 cups all-purpose flour and 1/4 cup cornstarch. Sift 4 times.