01 - Preheat the oven to 350°F. Grease and line a quarter sheet pan (9x13 inch) with parchment paper.
02 - In a bowl, whisk together cake flour, baking powder, baking soda, and salt until well combined.
03 - In a stand mixer, beat butter, shortening, and sugars on medium-high speed until light and fluffy, 2 to 3 minutes. Scrape down the bowl.
04 - Add eggs one at a time, beating well after each addition. Reduce speed; add buttermilk, milk, and vanilla extract. Mix until combined.
05 - On low speed, add dry ingredients and mix just until incorporated. Gently fold in rainbow sprinkles.
06 - Pour batter into prepared pan and smooth the top. Bake 28 to 32 minutes until a toothpick inserted comes out clean. Cool completely in the pan.
07 - Lower oven to 300°F. In a bowl, combine flour, sugars, baking powder, salt, and sprinkles. Add melted butter and vanilla; mix with a fork until small clusters form.
08 - Spread crumb mixture on a lined baking sheet and bake 15 minutes, stirring halfway through. Cool completely.
09 - In a stand mixer, beat butter, shortening, and cream cheese until smooth. Add vanilla and salt. Gradually beat in powdered sugar until fluffy. Add milk as needed for spreadable consistency.
10 - Mix milk and vanilla extract in a small bowl until combined.
11 - Using a 6-inch cake ring, cut two full circles and two half circles from the cooled cake sheet.
12 - Place a cake round inside a lined 6-inch cake ring set on a cake board. Brush with vanilla milk. Spread one-fifth of frosting over the cake and sprinkle one-third of the crumbs. Repeat layering with half-circles to fill in, milk soak, frosting, and crumbs.
13 - Top with final cake round, soak with vanilla milk, and finish with remaining frosting and crumb-sprinkle mixture.
14 - Chill cake in the ring for at least 1 hour, up to overnight. Remove ring, transfer to a platter, and serve at room temperature.