Preheat oven to 350 degrees F. Spray a 9x13-inch baking pan with nonstick spray.
In a large bowl, mix ricotta cheese, mascarpone cheese, vanilla extract and confectioners sugar until well combined and smooth. Transfer to a piping bag fitted with a large star tip. Refrigerate until ready to use.
Prepare cake according to package directions. Bake and let cool completely.
Use the flat tip of a large skewer to poke holes into the cooled cake, the more holes the better. Pour sweetened condensed milk over the cake and let soak for at least 30 minutes and up to an hour. Pipe the cannoli icing into stars over the top of the cake, covering completely. Sprinkle mini chocolate chips over the top to finish.