Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch baking pan.
Blend cake mix, 3 eggs, milk, and softened butter in a large bowl with an electric mixer on low speed until moistened, about 30 seconds. Increase speed to medium and beat until batter is smooth, about 4 minutes. Pour into prepared pan.
Beat remaining 2 eggs, ricotta cheese, sugar, and vanilla extract together in a large bowl until smooth. Pour evenly over cake batter in the pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 1 hour.
Chill cake in the refrigerator for at least 2 hours before serving.