Birthday Cake Delight

Name: Birthday Cake Delight
Prep time: 20 min
Total time: 20 min
Cuisine: American
Ingredients: 14.3 ounces vanilla sandwich cookies, ((approximately 36 cookies, 2 ½ cups when crushed)); 6 tablespoons salted butter, (melted and cooled (plus more for greasing dish)); 16 ounces cream cheese, (softened to room temperature); ¾ cup powdered sugar; 1 teaspoon vanilla extract; 16 ounces frozen whipped topping, (Cool Whip, defrosted and divided into 1 cup, 1 cup, and remainder); ¼ cup rainbow sprinkles, (plus more for garnish); 6.8 ounces vanilla-flavored instant pudding mix, (or white chocolate pudding); 2 cups whole milk