1. Heat oven to 350 degrees F. Coat a 13 x 9 x 2-inch baking dish with nonstick cooking spray.
2. In a medium saucepan, whisk flour and 1 cup of the milk until smooth. Stir in remaining 2 1/2 cups milk and bring to a boil; boil for 3 minutes, whisking constantly, until thickened. Whisk in salt, pepper, nutmeg and 1/2 cup of the Parmesan. Set aside.
3. Meanwhile, bring a large pot of lightly salted water to a boil. Add broccoli and cook 4 minutes; add mushrooms during last minute of cooking time. Using a slotted spoon, remove broccoli and mushrooms to a bowl. In the same pot, cook lasagna noodles following package directions, about 10 minutes. Drain.
4. Assemble lasagna: Spread 1/2 cup of the cheese sauce in bottom of prepared dish. Place 3 noodles over sauce, evenly spread ricotta mixture over top and add another layer of noodles. Spoon on vegetables, 2 cups of the sauce and 1/2 cup of the mozzarella. Place another layer of noodles over vegetables and top with the remaining 1 1/2 cups of cheese and remaining noodles. Evenly spread the remaining 1 cup of sauce over top and sprinkle with the remaining 1/4 cup Parmesan cheese.
5. Cover with nonstick foil and bake at 350 degrees F for 30 minutes; uncover and bake for an additional 15 minutes. Allow to stand 10 minutes before serving.