Birthday Cake

Name: Birthday Cake
Prep time: 2 h
Cook time: 30 min
Total time: 2 h 30 min
Cuisine: American
Ingredients: 1 cup unsalted butter (melted); 1 1/4 cup granulated sugar; 1/4 cup brown sugar; 3 large eggs (room temperature); 1 tablespoon vanilla extract; 1/2 cup buttermilk (substitute whole milk); 2 1/2 cups cake flour (substitute all purpose flour); 1 1/2 teaspoons baking powder; 1/2 teaspoon salt; 1/2 cup rainbow sprinkles; 1/4 cup unsalted butter (melted); 3/4 cup cake flour (substitute all purpose flour); 1/2 cup granulated sugar; 1/4 teaspoon baking powder; 2 tablespoons rainbow sprinkles; 1 teaspoons vanilla extract; 8 ounces cream cheese (room temperature); 1/2 cup unsalted butter (room temperature ); 2 cups powdered sugar; 1/4 teaspoon salt; 1 teaspoon vanilla extract; 2-3 teaspoons milk (as needed for desired consistency); 1/4 cup milk