Preheat oven to 350°F. Spray a 13 x 9-inch baking pan with nonstick spray.
Whisk together cake flour, baking powder, baking soda, and salt in a medium bowl. Set aside.
Beat butter and sugar in a large mixer bowl at medium speed until light and fluffy, then add vanilla. Add eggs one at a time and beat until combined.
Combine buttermilk and Greek yogurt in a small bowl and beat on medium-high. Reduce speed to low and add flour mixture in 3 additions, alternating with the buttermilk mixture (begin and end with flour mixture).
Spoon batter into prepared baking pan and smooth with an offset spatula. Tap pan sharply to reduce air bubbles and top batter with sliced strawberries.
Bake for 25 to 28 minutes, until the top and edges are golden, and a toothpick inserted in the center comes out clean or with a few crumbs attached.
Cool cake completely in the pan on a wire rack.
For the frosting:
For the frosting, beat butter, cream cheese, and salt in a large mixer bowl on medium speed until smooth. Beat in vanilla.
Mix in powdered sugar on low speed 1 cup at a time until incorporated, scraping down bowl as needed.
Increase speed to medium-high and beat frosting for 1 minute, until light and airy.
Dollop frosting over cooled cake and swirl with an offset spatula.
Chill cake for 20 minutes in the refrigerator before slicing and serving topped with fresh strawberries.
To make your own cake flour: For every 1 cup of cake flour, place 2 tablespoons cornstarch in an empty 1-cup measure, then top up with all-purpose flour. Sift the mixture several times to evenly distribute the cornstarch.