In a heavy saucepan, melt butter over medium heat. Stir in flour, salt and pepper until smooth. Gradually add milk. Bring to a boil; cook and stir for 2 minutes or until thickened. Remove from the heat; stir in onions, lemon juice and hot pepper sauce.
Spread a fourth of the white sauce in a greased 13x9x2-inch baking dish. Top with 3 noodles, half of the ham and broccoli, 3 T. Parmesan cheese, 1 cup cheddar cheese, half of the eggs and a 1/4 of the white sauce. Repeat layers.
Top with the remaining noodles, white sauce and cheeses. Bake, uncovered at 350°F for 40-45 minutes or until bubbly. Let stand 15 minutes before cutting.