Put the vegetables in a baking dish lined with baking paper to prevent them from sticking.
Drizzle with olive oil and add the garlic and thyme, season with salt and pepper.Use your hands to mix well then, roast for 20-30 minutes. Set aside.Bring a large pot of salted water to a boil over high heat.
Add the noodles and cook until al dente, 8 to 10 minutes.
Drain and rinse with cold water to cool.
Drain again and lay the noodles on a baking sheet and cover with a wet towel.In a saucepan over medium heat, bring the milk to just below a boil. In another saucepan over medium-high heat, melt the butter.Stir the flour into the butter and cook, stirring, for about 2 minutes.
Remove from the heat and gradually whisk in the hot milk.Set over medium-low heat and cook, stirring, until thick and smooth, about 3 minutes. Season with salt, pepper and add the nutmeg.
Remove the sauce from the heat. Set aside.Preheat oven to 375°F. Grease a 9-by-13-inch baking dish with olive oil.Cover the bottom of the prepared dish with a layer of the noodles.Spoon one-third of the tomato sauce over the noodles. Top with one-third of the roasted vegetables and then with one-third of the white sauce.
Sprinkle the mozzarella and asiago cheese evenly over the white sauce. Repeat the layering twice.
Bake until golden and bubbling, about 40 to 45 minutes.
Let stand for 15 minutes, then cut into squares and serve.
