Coat a 9- x 13-inch baking dish with cooking spray. Add batter and bake according to package directions. Allow for cake to cool.
Using the end of a wooden spoon (or straw or something similar in size), poke holes about 1 inch apart all over top of cake. Reserve 2 tablespoons condensed milk in a small bowl; pour remainder over top of cake and into holes.
In a large bowl, beat mascarpone, ricotta cheese, and vanilla until creamy. Beat in confectioner’ sugar and reserved condensed milk until smooth. Spread frosting evenly over top of cake. Sprinkle with chocolate chips.
Refrigerate cake 2 hours, or until ready to serve.