Preheat your oven to 350°F and prepare a 9×13-inch baking pan by greasing it and lining it with parchment paper strips. Make sure the parchment covers the bottom and the two longest sides of the pan, leaving some overhang for easy removal later. To prepare the strawberry puree, place the strawberries in a food processor and pulse just enough to almost completely puree them, leaving a few small chunks for texture. Use the same method to prepare the strawberries for the cream cheese frosting. In a bowl, whisk together the all-purpose flour, cake flour, baking powder, baking soda, and salt, then set this mixture aside. In a large mixing bowl, whisk the vegetable oil, sugar, eggs, strawberry puree, and vanilla extract until combined. Gently fold in the dry ingredients until the batter just comes together. Pour the batter evenly into the prepared pan and smooth the top. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean with a few moist crumbs attached. Allow the cake to cool completely in the pan before frosting. If you want to remove the cake from the pan, refrigerate it for at least one hour to firm it up.

Prepare the frosting as soon as the cake finishes baking so it can chill alongside the cake. Using a hand mixer or stand mixer fitted with the paddle attachment, beat together the cream cheese and butter until smooth. Gradually add the powdered sugar, about one cup at a time, mixing well after each addition. Then, mix in the strawberry puree until fully combined. Refrigerate the frosting for about an hour to let it thicken. Once the cake has completely cooled, spread the frosting evenly over the top. Serve immediately or store the frosted cake in the refrigerator for up to seven days. Enjoy!