Preheat oven to 400 degrees. In a large saucepan, heat oil over medium; add flour and garlic. Cook, stirring constantly, 2 to 3 minutes (do not let flour mixture darken); whisk in milk. Bring to a boil; reduce to a simmer, and cook, whisking occasionally, until thickened, 3 to 5 minutes. Add spinach, peas, and carrots; season with salt and pepper. Set sauce aside.
In a medium bowl, combine ricotta, egg, 1/2 teaspoon salt, and teaspoon pepper.
In the bottom of a 9-by-13-inch baking dish, spread a thin layer of vegetable sauce. Layer noodles, half the remaining vegetable sauce, another noodles, half the ricotta mixture, half the mozzarella, and half the Parmesan; repeat.
Cover dish with lightly oiled aluminum foil, and place on a large rimmed baking sheet. Bake 45 minutes, uncover, and bake until bubbling and browned, about 20 minutes more. Let cool 10 to 15 minutes before serving.
To freeze: Prepare through step 3. Cover lasagna tightly with plastic wrap, then aluminum foil; freeze up to 3 months.
To bake from frozen: Remove plastic wrap; cover baking dish with lightly oiled aluminum foil, and place on a large rimmed baking sheet. Bake at 400 degrees for 1 1/2 hours. Remove foil, and bake until bubbling and browned, about 30 minutes more. Let cool 10 to 15 minutes before serving.