Preheat the oven to 350 degrees F. Grease and flour a 9-by-13-inch baking pan.
Prepare the cake batter according to the package directions. Spoon evenly into the prepared pan and set aside.
Combine the ricotta, sugar, cocoa powder, vanilla, eggs and 4 ounces of the mascarpone in a stand mixer fitted with the paddle attachment. Beat until smooth. Gently scoop the filling onto the cake batter and spread, so the top is completely covered. Bake until a skewer inserted into the center of the cake comes out clean and the vanilla cake layer has risen to the top, 50 to 60 minutes. Let cool completely in the pan on a cooling rack.
Combine the instant pudding, milk and the remaining 12 ounces mascarpone in a stand mixer and beat until thick and smooth. Spread the frosting evenly over the entire cake. Use a peeler to shave chocolate curls to garnish the cake.