Special equipment: parchment paper; three 8-inch round cake pans

Preheat the oven to 325 degrees F. Spray three 8-inch round pans with nonstick spray and line them with parchment paper. Spray the paper.

Sift together the all-purpose flour, cake flour, baking powder, baking soda and salt in a large bowl. Whisk together the egg whites, sour cream, milk, vanilla and vanilla bean seeds in another large bowl. 

Beat the butter and sugar with a paddle attachment until light and fluffy. Alternate beating in the flour mixture and the egg mixture, adding the flour mixture in 3 additions and the egg mixture in 2, starting and ending with the flour.

Fill each cake pan with 2 1/2 cups of batter and bake until a toothpick inserted in the center comes out clean, about 40 minutes. Cool the cakes in the pans for 15 minutes, then flip them onto a cooling rack to cool completely. 

For the filling: Put the strawberries, granulated sugar, balsamic vinegar and lemon juice in a medium saucepan over medium heat, bring the mixture to a simmer, and cook until it gets to a syrupy reduction, about 30 minutes. 

Cool completely. 

Beat the mascarpone in a medium bowl until smooth. Whip the heavy cream until light and fluffy in another bowl; just before it is finished, whip in the confectioners¿ sugar and vanillia extract. Fold in the strawberry mixture. 

For the frosting: Cream the butter and shortening until light and fluffy, then beat in the confectioners' sugar and vanilla extract. 

To assemble the cake: Put 1 cake layer on a cake board or platter. Spread half of the filling mixture on top. Repeat with another cake layer. Frost the cake with the buttercream frosting and serve.