In a food processor, pulse freeze-dried strawberries until finely ground; transfer to a small bowl and reserve for the frosting (this is so you don’t have to wipe out the food processor twice—you don’t need freeze-dried strawberries for the batter, just the frosting).

In food processor, pulse strawberries to a smooth puree (a few small chunks are okay). Transfer puree to a medium pot and bring to a boil over medium-high heat. Reduce heat to medium and simmer, stirring occasionally, until reduced by half, 10 to 15 minutes (you should have about 1 cup).

Pour puree into a heatproof container and let cool completely.

Meanwhile, arrange a rack in center of oven; preheat to 350°. Grease 2 (8”) round cake pans with cooking spray and line bottoms with parchment.

In a medium bowl, whisk flour, cornstarch, baking powder, and salt.

In a large bowl, using a handheld mixer on medium-high speed (or in the large bowl of a stand mixer fitted with the paddle attachment), beat granulated sugar and butter until light and fluffy, 2 to 3 minutes (mixture will be grainy). Add oil and beat until combined, about 1 minute more. Add egg whites, one at a time, beating to blend and scraping bowl after each addition, until incorporated. (Batter may start to look split by the third egg white; if this happens, add 1 to 2 Tbsp. of the dry ingredients, which will help the mixture emulsify again.)

Add half of dry ingredients and beat on low speed until just a few streaks remain. Add cooled strawberry puree and milk and beat just until combined. Scrape down bowl, then add remaining dry ingredients and fold with a spatula until combined. Divide batter between prepared pans.

Bake cake until it slightly pulls away from the sides of pan and lightly springs back when pressed in the center, 23-27 minutes. Let cakes cool completely, then remove from pans.

Using a fine-mesh sieve, sift confectioners’ sugar into a medium bowl. Add strawberry powder to sieve and sift over top.

In the large bowl of stand mixer fitted with the paddle attachment, beat butter and salt on medium-high speed until softened and creamy, 1 to 2 minutes. Add cream cheese and beat on medium speed until smooth and combined, about 1 minute more.

Spoon one-third of strawberry confectioners’ sugar into butter mixture and beat on low speed, scraping down bowl as needed, until incorporated. Add another third of strawberry sugar and beat on low speed until incorporated, then repeat with remaining strawberry sugar. Add heavy cream and beat until combined. Increase mixer speed to medium and continue to beat until light and fluffy, 1 to 2 minutes more.

To assemble cake, using a serrated knife, level the flattest cake layer. Arrange on a platter. Spread with 1 cup buttercream. Place second cake layer on top. Frost top and sides of cake with remaining buttercream as desired.