1. For the cake: Adjust oven rack to middle position and heat oven to 350 degrees. Grease two 9-inch round cake pans, line bottoms with parchment, grease parchment, and flour.
2. Transfer strawberries to bowl, cover, and microwave until strawberries are soft and have released their juice, about 5 minutes. Place in fine-mesh strainer set over small saucepan. Firmly press fruit dry (juice should measure at least 3/4 cup); reserve strawberry solids. Bring juice to boil over medium-high heat and cook, stirring occasionally, until syrupy and reduced to 1/4 cup, 6 to 8 minutes. Whisk milk into juice until combined.
3. Whisk strawberry milk, egg whites, and vanilla in bowl. Using stand mixer fitted with paddle, mix flour, sugar, baking powder, and salt on low speed until combined. Add butter, 1 piece at a time, and mix until only pea-size pieces remain, about 1 minute. Add half of milk mixture, increase speed to medium-high, and beat until light and fluffy, about 1 minute. Reduce speed to medium-low, add remaining milk mixture, and beat until incorporated, about 30 seconds. Give batter final stir by hand.
4. Scrape equal amounts of batter into prepared pans and bake until toothpick inserted in center comes out clean, 20 to 25 minutes, rotating pans halfway through baking. Cool cakes in pans on wire rack for 10 minutes. Remove cakes from pans, discarding parchment, and cool completely, about 2 hours. (Cooled cakes can be wrapped with plastic wrap and stored at room temperature for up to 2 days.).
5. For the frosting: Using stand mixer fitted with paddle, mix butter and sugar on low speed until combined, about 30 seconds. Increase speed to medium-high and beat until pale and fluffy, about 2 minutes. Add cream cheese, one piece at a time, and beat until incorporated, about 1 minute. Add reserved strawberry solids and salt and mix until combined, about 30 seconds. Refrigerate until ready to use, up to 2 days.
6. Pat strawberries dry with paper towels. When cakes are cooled, spread 3/4 cup frosting over 1 cake round. Press 1 cup strawberries in even layer over frosting and cover with additional 3/4 cup frosting. Top with second cake round and spread remaining frosting evenly over top and sides of cake. Garnish with remaining strawberries. Serve. (Cake can be refrigerated for 2 days. Bring to room temperature before serving.).