Preheat your oven to 350°F and grease a 9×13-inch baking pan. Line the pan with a strip of parchment paper, making sure it covers the bottom and the two longest sides, leaving some overhang for easy cake removal. To prepare the strawberry puree, place the strawberries in a food processor and pulse for a few seconds until mostly pureed but still with a few small chunks remaining. This same method is used for the strawberries in the cream cheese frosting. In a bowl, whisk together the all-purpose flour, cake flour, baking powder, baking soda, and salt, then set aside. In a large mixing bowl, combine the vegetable oil, sugar, eggs, strawberry puree, and vanilla extract, whisking until smooth. Gently fold the dry ingredients into the wet mixture until just combined. Pour the batter evenly into the prepared pan and smooth the surface. 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 prefer to remove the cake from the pan before frosting, refrigerate it for at least one hour to firm it up.

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