Preheat oven to 350°F and grease three 9-inch cake pans.
In a large bowl, whisk together the flour, baking powder and salt. Set aside.
With an immersion blender* puree the strawberries until smooth. In a small bowl, mix the strawberry puree with the milk and vanilla until combined. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and sugar on medium speed until smooth and creamy, scraping down the side of the bowl as necessary.
Reduce the speed to low and add the eggs and egg whites one at a time, adding each egg only when a streak of the former egg remains, and scraping down the side of the bowl as necessary.
While on the low speed add half of the flour mixture. Once mostly combined, add the strawberry puree mixture and continue mixing until combined. Scrape down the side and bottom of the bowl and add the remaining flour mixture. Mix until just a few streaks of flour remain. Remove bowl from the mixer and mix by hand using a rubber spatula until no flour streaks remain.
Divide the batter evenly among the three prepared cake pans. Bake for 22 to 25 minutes, until a toothpick inserted in the center comes out clean.
Allow to cool completely in the pan on a cooling rack. Run a thin knife or spatula along the rim of the pans and invert the pans to remove the cakes.
Frost with strawberry frosting.
Strawberry Frosting
With an immersion blender* puree the strawberries until smooth. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium-high speed until pale and creamy.
Reduce the speed to low and add the powdered sugar, 1/2 cup at a time. Scrape down the side and bottom of the bowl with a rubber spatula.
Measure out 6 tablespoons of the strawberry puree (most likely all of it) and add to the mixture, followed by the vanilla and salt. Continue mixing on low until combined. Increase the speed to medium for 10 to 15 seconds, until whipped and fluffy.
Notes
*Although I highly recommend an immersion blender like this one for the smoothest puree, a food processor can also be used.