Cake: 350? Rack in middle. Grease two 9" round pans, line bottoms with parchment, grease parchment, and flour.
Place strawberries in bowl, cover, and microwave until berries are soft and released their juice, about 5 min. Place in fine-mesh strainer set over small saucepan. Firmly press fruit dry (juice should measure at least 3/4 C); reserve strawberry solids. Bring juice to boil over med-high heat and cook, stirring occasionally, until syrupy and reduced to 1/4 C, 6 to 8 min.
Whisk strawberry milk, egg whites, and vanilla in bowl. Using stand mixer with paddle, mix flour, sugar, baking powder, and salt on low until combined. Add butter, 1 piece at a time, and mix until pea-size pieces, about 1 min. Add half of milk mixture, increase speed to med-high, and beat until light and fluffy, about 1 min. Reduce to med-low, add remaining milk mixture, and beat until incorporated, about 30 sec. Give batter final stir by hand.
Put batter into prepared pans and bake until toothpick inserted in center comes out clean, 20 to 25 min, rotating pans halfway through. Cool cakes in pans on wire rack for 10 min. Remove cakes from pans, and parchment. Cool completely, about 2 hrs.
Frosting: Using stand mixer with paddle, mix butter and sugar on low until combined, about 30 sec. Increase to med-high and beat until pale and fluffy, about 2 min. Add cream cheese, one piece at a time, and beat until incorporated, about 1 min. Add reserved strawberry solids and salt and mix until combined, about 30 sec.
Pat strawberries dry with paper towels. Spread 3/4 C frosting over 1 cake round. Press 1 C strawberries in even layer over frosting and cover with additional 3/4 C frosting. Top with second cake and spread remaining frosting evenly over top and sides of cake. Serve room tempature.