Hull and puree strawberries in a food processor or blender until completely smooth. You need 1.5 cups puree; strain through a fine-mesh sieve if you want to remove seeds (optional). Preheat oven to 350°F. Line a half sheet pan with parchment paper, leaving overhang on two sides for easy removal. Lightly grease exposed sides.

In a medium bowl, whisk together flour, sugar, baking powder, baking soda, and salt until well combined. The sugar gets mixed with dry ingredients here for a tender crumb.

In a large bowl with an electric mixer, beat softened butter on medium speed until creamy, about 2 minutes. Add eggs one at a time, beating well after each. Mix in sour cream and vanilla until smooth. Add dry ingredients in three additions, alternating with strawberry puree in two additions, beginning and ending with dry. Mix just until combined after each addition; do not overmix. Batter will be thick and pink.

Spread batter evenly in prepared pan with an offset spatula. Bake 30-35 minutes, rotating pan halfway through, until cake springs back when lightly touched and a toothpick inserted in center comes out clean. Let cool completely in pan on a wire rack, about 45 minutes.

In a large bowl, beat cream cheese and butter together on medium speed until completely smooth and fluffy, about 3 minutes. Scrape down bowl twice to eliminate lumps.

Add powdered sugar, vanilla, and salt. Beat on low until incorporated, then increase to medium-high and beat until light and fluffy, about 2 minutes. Spread frosting over completely cooled cake. For clean slices, refrigerate 20 minutes before cutting. Store covered in refrigerator up to 3 days.