Preheat your oven to 350°F (175°C). Cut the cinnamon rolls into 1-inch pieces and grease a 9x13-inch baking dish. Place the cinnamon roll pieces evenly in the dish.

In a mixing bowl, whisk together the eggs until fluffy. Add milk, cream cheese, brown sugar, and vanilla extract, whisking until smooth.

Pour the egg mixture over the cinnamon roll pieces, ensuring each piece is coated. Toss gently to thoroughly immerse the bites in the liquid.

Bake the casserole in the preheated oven for about 30 minutes, until golden brown and puffy. A toothpick inserted in the center should come out clean.

Allow the casserole to cool for 10 minutes. Prepare the icing by whisking powdered sugar, milk, and vanilla until smooth. Drizzle over the casserole before serving.