In a bowl, combine eggs, milk, salt, and pepper; mix well. Stir in cooked rice and sliced canned mushrooms.
Pour egg and rice mixture into a greased 8 x 8-inch baking dish.
Bake at 350° F for 30 to 35 minutes, or until the center is set.
When ready to serve, melt butter in a heavy saucepan over low heat. Add flour and stir until smooth. Cook for 1 minute, stirring constantly. Gradually add milk and cook over medium heat, stirring constantly, until thick and bubbly. Add shredded cheddar cheese, salt, and pepper; stir until the cheese melts then serve over the eggs.Servings: 8