Prep the Vegetables: In a large skillet, heat the olive oil over medium heat. Add the onion and garlic, and sauté until translucent. Add zucchini, yellow squash, red bell pepper, and broccoli. Cook until vegetables are tender. Stir in spinach until wilted. Set aside to cool.
Prepare the White Sauce: In a saucepan, melt butter over medium heat. Stir in flour to create a roux. Gradually add milk, whisking continuously until the sauce thickens. Season with salt, pepper, and a pinch of nutmeg.
Cook the Noodles: Bring a large pot of salted water to boil. Cook the lasagna noodles according to package instructions until al dente. Drain and lay flat on a sheet to prevent sticking.
Assemble the Lasagna: In a mixing bowl, combine ricotta cheese with the egg. In a greased 9x13 inch baking dish, layer as follows: a thin layer of white sauce, a layer of noodles, half the ricotta mixture, half the vegetable mixture, a third of the mozzarella, and a third of the sauce. Repeat the layers and top with remaining noodles, sauce, mozzarella, and Parmesan cheese.
Bake: Preheat the oven to 375°F (190°C). Cover the lasagna with aluminum foil and bake for 25 minutes. Remove the foil and bake for an additional 15 minutes, or until the cheese is bubbly and golden. Let stand for 10 minutes before serving.