Apple and Browned Butter Baked Oatmeal

Name: Apple and Browned Butter Baked Oatmeal
Cuisine: American
Ingredients: 4 large apples, peeled, cored and diced; 8 tablespoons unsalted butter, divided, plus more for baking; 1/2 cup maple syrup, divided; 4 1/2 cups old-fashioned rolled oats; 1 tablespoon baking powder; 1/2 teaspoon salt; 1 1/2 teaspoons ground cinnamon; 3 large eggs; 2 cups milk, plus more for serving; 1 teaspoon pure vanilla extract