Place the cream cheese, egg, egg yolk, cake mix, sugar, and vanilla in a medium bowl. Whisk until smooth, scraping down the sides of the bowl as needed. Mix in your rainbow sprinkles and set aside.
Place your remaining brownie batter over the birthday layer, then carefully spread it out. Use a butter knife to swirl the batter and the filling together slightly.
Bake your brownies for 30-35 minutes, or until a toothpick inserted into the center comes out just clean, it should have some crumbs. The center can still be a little jiggly.