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