Lightly grease a 9x13-inch pan with nonstick spray. Heat oven to 350°F.
In a medium bowl, whisk the flour, baking powder, baking soda, and salt to combine.
In the bowl of an electric mixer with the paddle attachment, cream the butter and sugar on medium speed until light and fluffy, 4 to 5 minutes.
Beat in eggs one at a time, then the egg yolk, scraping the sides of the bowl after each addition.
Mix on medium speed until fully incorporated, about 1 minute. Add the vanilla extract and mix to combine, 15 to 30 seconds.
Add 1/3 of the dry ingredients and mix on low until incorporated, about 30 seconds. Add half the milk and mix on low for 15 seconds. Repeat with another batch of flour mixture, the remaining milk, then the remaining flour mixture until combined.
Transfer batter to the prepared pan and spread evenly. Bake until a toothpick inserted into the center comes out clean and cake is golden, 30 to 35 minutes. Cool completely.
Purée the strawberries in a food processor.
Cream butter and strawberry purée in mixer for about 3 minutes.
Add powdered sugar a little at a time on low speed. Once incorporated, increase to high speed until light and fluffy 30-45 seconds.
Cut the cake in half using a serrated knife. Use 1/2 of the frosting to fill the cake. Layer with strawberries. Then, stack and frost the top.