Step 1

Preheat oven to 425ºF. In a large bowl, toss vegetables with 3 tablespoons olive oil and 1 teaspoon salt. Spread out on a rimmed baking sheet and roast in the center of the oven for 25 minutes. Toss and return to the oven for 10-15 minutes longer, or until onions are starting to brown.

Step 2

While the veggies are roasting, heat 3 tablespoons oil with garlic and red pepper flakes, in a large cooking pot or dutch oven over medium heat. When the garlic starts to sizzle, let it cook, stirring constantly, for 30-45 seconds (don't let it brown). Pour in the tomatoes, tomato paste, water, basil and 1 teaspoon salt. Bring to simmer and cook, partially covered, for 30 minutes. Puree the sauce with an immersion blender or in a food processor or blender, in batches. Stir in shaved broccoli and roasted veggies.

Step 3

While the sauce is cooking, bring a large pot of water to a boil. Add 1 tablespoon olive oil and 1 teaspoon of salt to the water. Lower the lasagna noodles into the pot one at a time so they're less likely to stick together. Cook until al dente, per package instructions, stirring often with a wooden spoon to separate noodles. Drain noodles, run some cool water over them, and lay them out on a sheet pan. Cover with a damp dish towel.

Step 4

Preheat the oven to 375ºF and set the oven rack in the center position. Spread a little sauce over the bottom of the casserole dish. Arrange a layer of 4-5 noodles on top. Cover with ⅓ of the sauce. Sprinkle on ⅓ of the mozzarella. Spoon ½ of the ricotta around evenly, in little blobs. Sprinkle on ⅓ of the parmigiana cheese. Repeat with a second layer of noodles, half the remaining sauce, half the remaining mozzarella, the rest of the ricotta, half of the remaining parmigiana. Add the final layer of noodles, the rest of the sauce, the rest of the mozzarella and parmigiana.

Step 5

Cover the casserole tightly with a double layer of foil but tent it a bit so the foil doesn't touch the top layer of cheese. Cook for 40 minutes. Remove the foil and cook for 10-15 minutes more, until the sauce is bubbly and the cheese is melted. Let the lasagna rest for 15-20 minutes. Slice and serve. Sprinkle with fresh basil, if you like.