Preheat oven to 350 degrees F. Lightly grease a 3-quart rectangular baking dish; set aside. Cook lasagna noodles according to package directions; drain. Rinse with cold water; drain again. Place noodles in a single layer on a sheet of foil; set aside.
Meanwhile, for sauce, in a medium saucepan, whisk together about 1 cup of the milk and the flour until smooth. Stir in the remaining milk, salt, dry mustard, black pepper, and hot pepper sauce. Cook and stir over medium heat until thickened and bubbly. Remove from heat; stir in Parmesan cheese.
Arrange three of the cooked lasagna noodles in the prepared baking dish. Top with half of the broccoli, half of the carrots, 3/4 cup of the cheddar cheese, and 1 cup of the sauce. Repeat layering noodles, broccoli, carrots, cheese, and sauce. Top with the remaining noodles, sauce, and cheese.
Bake, uncovered, for 35 to 40 minutes or until heated through. Let stand for 15 minutes before serving. Makes 8 servings.