Preheat your oven to 350°F. Line a baking sheet with parchment paper and set it aside. In the bowl of a stand mixer fitted with the paddle attachment, combine all the crumble ingredients except for the butter and mix on medium-low speed until well combined. Add in the softened butter and continue mixing on low until the mixture forms clumps and is fully incorporated. Spread the crumble mixture evenly over the prepared baking sheet. Bake for 5 minutes, then use a spatula to stir and redistribute the crumble. Bake for an additional 5 minutes, or until it develops a light golden color. Let it cool completely, then store in an airtight container at room temperature until ready to use. The crumble will keep fresh for up to 5 days.

Preheat the oven to 325°F. Prepare three 8-inch or four 6-inch round cake pans by spraying them with nonstick spray, lining the bottoms with parchment paper, and spraying again. Set aside. In a medium mixing bowl, whisk together the cake flour, baking powder, baking soda, and salt. In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter, oil, and sugar on high speed for about 4 minutes, until light and fluffy. With the mixer on medium speed, add the whole egg, followed by the egg whites one at a time, scraping down the sides and bottom of the bowl between additions. After the eggs are fully incorporated, beat on medium speed for an additional minute. Add the sour cream and vanilla, and mix on medium for another 30 seconds. Reduce the mixer speed to low, then add half of the dry ingredients and half of the buttermilk, mixing until just incorporated. Add the remaining dry ingredients and buttermilk and continue mixing until smooth. Gently fold the flour-coated berries into the batter using a rubber spatula. Divide the batter evenly between the pans using a kitchen scale (approximately 18 oz per 8-inch pan or 12 oz per 6-inch pan). Bake for 25–30 minutes, or until the tops are lightly golden and a toothpick inserted in the center comes out with just a few moist crumbs. Let the cakes cool in their pans for 15 minutes before removing them and placing them on a wire rack to cool completely. Once cool, wrap each layer in plastic wrap and refrigerate or freeze until ready to assemble. If needed, level any domed tops before chilling.

In a small saucepan over medium heat, combine the mixed berries, sugar, and ¾ cup of water. Stir together and cook until the berries start to break down and the mixture becomes sauce-like. In a separate small bowl, mix the cornstarch with a bit of water to create a slurry. Once the berry mixture is bubbling, stir in the slurry. The compote will initially turn light purple, but continue cooking until it thickens and darkens—this ensures the cornstarch flavor is fully cooked out. Remove from heat and let cool to room temperature. The compote will continue to thicken as it cools. To prevent a skin from forming, press a layer of plastic wrap directly on top. You may refrigerate the compote to speed up the cooling process if needed.

In the bowl of a stand mixer fitted with a paddle attachment, beat the butter on medium speed for about 2 minutes until creamy. Scrape down the bowl. Add the sifted powdered sugar and mix on low speed until incorporated. Scrape the sides and bottom of the bowl again, then increase the mixer to medium and add in the heavy cream, vanilla bean paste, cinnamon, and salt. Beat on medium-high speed for 5 minutes until the buttercream is fluffy and pale. In a separate bowl, combine about 2 cups of the prepared vanilla buttercream with ¼ cup of the cooled berry compote. Stir gently with a spatula until fully mixed. Before frosting the cake, spend a few minutes mixing the buttercream by hand with a spatula or wooden spoon to eliminate any air bubbles and create a smooth, silky texture.

Place a small amount of frosting in the center of a cake board to secure the first layer. Position the first cake layer top side up. Spread half of the berry-infused buttercream over the top, smoothing it evenly to the edges. Pipe a ring of plain vanilla buttercream around the edge to act as a barrier for the filling. Spoon about ⅓ cup of the berry compote inside the frosting ring, being careful not to overfill. Sprinkle half of the oatmeal crumble over the compote layer. Gently add the second cake layer on top and repeat the process: berry buttercream, frosting dam, compote, and crumble. Place the final cake layer top side down on the second layer of filling. Spread a thin layer of buttercream around the entire cake to create a crumb coat and seal in any loose crumbs. Freeze the cake for about 10 minutes to set the crumb coat. Once chilled, apply the remaining buttercream to frost and decorate the cake as desired.