Preheat the oven to 350 degrees F. Spray a 9-by-13-inch baking pan with cooking spray.
Prepare the cake according to the directions on the box and pour it into the prepared pan. Set aside.
Combine the ricotta, sugar, vanilla, eggs and 4 ounces (1/2 cup) of the mascarpone in the bowl of a stand mixer fitted with a paddle attachment. Beat on medium speed until smooth, scraping down the sides of the bowl with a rubber spatula as needed. Gently scoop the filling onto the cake batter and cover it completely.
Bake until a skewer inserted in the center of the cake comes out clean and the strawberry layer has risen to the top, 50 to 60 minutes. Cool completely on a rack.
Combine the remaining 12 ounces mascarpone with the pudding mix and milk in a clean bowl in the stand mixer fitted with a paddle attachment. Beat on low speed until incorporated. Increase the speed to medium and beat until smooth and creamy. Spread the frosting evenly over the cake. Serve.