Preheat the oven to 350°F (175°C). Spray a 9x13-inch baking dish with cooking spray.
Prepare the red velvet cake batter according to the package instructions, using the additional ingredients listed on the box.
Pour the prepared red velvet batter into the greased 9x13-inch baking dish and spread evenly.
In a mixing bowl, use a hand mixer to beat the ricotta, eggs, sugar, and vanilla until smooth and well combined.
Carefully spread the ricotta mixture over the red velvet batter in the pan, smoothing it gently with a spatula.
Bake for about 1 hour, or until the cake is set and a toothpick inserted into the center comes out mostly clean.
Remove the cake from the oven and let it cool completely in the pan on a wire rack before frosting.
For the topping, beat the instant cheesecake pudding mix, softened cream cheese, and cold milk with a hand mixer until smooth.
Gently fold the Cool Whip into the pudding–cream cheese mixture until evenly combined.
Spread the pudding and Cool Whip mixture over the completely cooled cake, then refrigerate the cake for a few hours before serving.