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Preheat oven to 350°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, whisk 1 cup milk and the flour in a medium saucepan until smooth. Stir in the remaining 2 1/2 cups milk, salt, dry mustard, black pepper and hot sauce. Cook and stir over medium heat until thickened and bubbly. Remove from heat; stir in Parmesan.
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 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.