Preheat the oven to 350 degrees.
In a food processor, puree strawberries. Add to a small saucepan over medium low heat with sugar and lemon juice. Cook, stirring often, until thickened and reduced by about half, about 20 minutes. Remove from heat and let cool completely.
Grease and line two8-inch round cake pans. In a large bowl, whisk together flour, cornstarch, baking powder, baking soda, and salt.
In another large bowl, with a hand mixer, beat together butter and sugar until light and fluffy.Add eggs, one at a time, beating well after each addition. Beat in strawberry compote and vanilla until fully incorporated, then add milk. Add dry ingredients, mixing until just combined.
Divide batter evenly into prepared pans and bake until a toothpick inserted into the middle of each comes out clean, 25 minutes. Let cakes cool for 15 minutes then invert onto cooling racks to cool completely.
For the frosting, use a hand mixer to the beat cream cheese and butter until no lumps remain. Beat in strawberry compote. Add powdered sugar and beat until fully incorporated. Add vanilla and a pinch of salt.
Spread a thick layer of frosting on top of one cake layer. Top with the second layer of cake, then frost all over the sides and top. Garnish with fresh strawberries.