For the puree: Pulse the strawberries in a food processor, scraping as needed, until a smooth puree forms. Add the strawberry puree to a small saucepan. Bring to a boil over medium heat. Reduce the heat to low and simmer until the puree reduces by half (to about 1 cup). Transfer the puree to a bowl and set aside to cool.
For the cake: Position a rack in the center of the oven and preheat to 350 degrees F. Lightly coat two 9-inch round cake pans with nonstick baking spray. Line the bottoms with parchment rounds; lightly coat parchment with nonstick spray.
Whisk together the flour, cornstarch, baking powder, salt and baking soda in a large bowl until combined. Beat the butter, granulated sugar, oil and vanilla in another bowl with an electric mixer on medium-high speed until light and fluffy, 3 to 4 minutes. Add the whole eggs and yolks one at a time, beating to blend between additions and occasionally scraping down the sides and bottom of the bowl, until the batter is completely combined, smooth and creamy. With the mixer on low speed, add the dry ingredients in 3 additions, alternating with the buttermilk, beginning and ending with the dry ingredients. Fold in 1/2 cup of the cooled strawberry puree. Reserve the rest of the puree for the frosting.
Scrape the batter evenly into the prepared pans. Smooth the tops. Bake, rotating the pans halfway through, until the cakes are a light golden brown and a tester inserted into the centers comes out clean, 35 to 45 minutes. Transfer the pans to a wire rack and let the cakes cool for 30 minutes. Run a knife around the sides of the cakes and invert onto the wire rack. Remove the parchment and let the cakes cool completely.
For the frosting: Beat the cream cheese in a large bowl with an electric mixer on medium speed until smooth and fluffy. Gradually beat in the butter, beating until smooth. Sift the confectioners' sugar into the bowl and beat until smooth. Add the vanilla and beat until light and fluffy. Remove 3 cups of the frosting to a medium bowl and set aside. Fold the reserved strawberry puree into the remaining frosting. Refrigerate both until slightly chilled and firm, about 20 minutes.
To assemble the cake: Place one cake domed-side down on a platter. Spread 1 1/2 cups of the strawberry frosting evenly over the top. Place the remaining cake domed-side up on top. Spread the top and sides of the cake with 1 1/2 cups of the white frosting. Smooth out completely to make an even crumb coat. (It's okay if some of the cake shows through. This is just a base layer.) Refrigerate for 30 minutes to let the frosting set.
Spread the remaining white frosting on the top of the cake and the upper half of the sides. There should be a band all around the bottom of the cake with the crumb coat exposed. Fill in the exposed area with 1 cup of the strawberry frosting. Use a large offset spatula to smooth the top and sides of the cake. The strawberry and white frostings should blend together during smoothing to create an ombre effect. Transfer the remaining strawberry frosting to a piping bag or resealable plastic bag fitted with a medium star tip. Pipe 8 rosettes evenly spaced along the top border of the cake.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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