Preheat your oven to 400°F (200°C). On a baking sheet, toss chopped broccoli, cauliflower, carrots, peas, and beets with olive oil, salt, and pepper. Spread the vegetables out in a single layer and roast for about 30 minutes, or until they are tender and slightly charred.

While the vegetables are roasting, bring a large pot of salted water to a boil. Cook the lasagna noodles according to package directions, usually around 8-10 minutes, until al dente. Drain the noodles and lay them flat on a clean kitchen towel to prevent sticking.

In a mixing bowl, combine ricotta cheese, a beaten egg, shredded mozzarella, chopped spinach, and a pinch of salt and pepper. Mix these ingredients until fully incorporated.

In a saucepan over medium heat, melt 2 tablespoons of butter. Add chopped shallots and sauté until translucent, about 3 minutes. Stir in 2 tablespoons of flour and cook for another minute, creating a roux. Gradually whisk in heavy cream, along with rosemary, thyme, garlic powder, salt, and pepper. Allow the sauce to thicken, stirring frequently for about 5-7 minutes before mixing in grated Parmesan cheese.

Lower the oven temperature to 350°F (175°C). In a small baking dish, spread a thin layer of béchamel sauce along the bottom. Place a layer of cooked lasagna noodles over the sauce, followed by spoonfuls of the ricotta mixture and a generous layer of roasted vegetables. Repeat these layers, finishing with a final layer of sauce and a sprinkle of shredded mozzarella on top.

Cover the assembled lasagna with foil and bake in the preheated oven for 35 minutes. Then, remove the foil and bake for an additional 5-10 minutes, or until the top is golden and bubbling. Let the lasagna rest for about 15 minutes before slicing into it.