Preheat the oven to 400°F. Position the racks in the upper and lower thirds of the oven. Line two large baking sheets with parchment paper.
Prepare the tomato sauce: Place the tomatoes and onion on one of the baking sheets. Drizzle with 2 tablespoons of the olive oil, balsamic vinegar, and season with the salt, oregano, and pepper to taste. Toss well to coat, then spread out in a single layer. Prepare the garlic by using a sharp knife, cut ¼-inch off from the stem end of the head so that the individual cloves of garlic are exposed. Place the garlic in a square piece of foil, drizzle with the remaining 1 tablespoon of olive oil, then loosely wrap in the foil and place on the sheet pan with the tomatoes and onions. Set aside while you prep your veggies.
Prepare the veggies: On the other large baking sheet, combine the bell peppers, onion, and zucchini, drizzle with the olive oil, and season with the garlic powder, salt, and pepper to taste. Use your hands to toss everything together, then spread the veggies out in an even layer. (If you don’t have two large baking sheets you can saute all of the veggies in a skillet with a few tablespoons of olive oil for 5 to 8 minutes until tender.)
Roast: Place both sheet pans in the oven on two different racks. Rotate the pans halfway through the baking time. Roast the veggies, flipping halfway through, until they are slightly golden, 30 to 35 minutes. Set aside to cool. Keep the tomatoes, onion, and garlic in the oven, undisturbed, for 10 to 15 minutes longer, until the tomatoes are tender and juicy and the onions are soft, a total of about 45 minutes. Keep the oven on and carefully position a rack in the middle. Keep heat in the oven.
Bring a large pot of generously salted water to a boil.
Cool the tomato mixture for 10 minutes, then remove the head of garlic from the foil packet and discard the foil. Once the garlic is cool enough to handle, use your fingers to squeeze the roasted garlic from the skin directly into a high-powered blender. Add the roasted cherry tomatoes and onions, along with their juices, to the blender and blend on medium-low until a smooth sauce forms, 1 to 2 minutes. Set aside.
Cook the noodles: Spray a baking sheet or cutting board with cooking spray or line with parchment paper. Cook the lasagna noodles for 5 to 6 minutes, until pliable but still slightly firm, then drain and immediately lay the noodles flat in a single layer on the prepared baking sheet or cutting board. Do not let the lasagna noodles touch or they’ll stick together.
Make the pesto ricotta: In a medium bowl, mix the ricotta, pesto, egg, salt, garlic powder, and pepper to taste, until smooth.
Assemble the lasagna: Grease a 9 × 13-inch baking dish with cooking spray. Spread 1 cup of the tomato sauce over the bottom. Place 5 of the cooked lasagna noodles on top, laying 4 horizontally and 1 vertically to cover the bottom. Spread half of the pesto-ricotta mixture on top of the noodles, followed by half of the roasted veggies, and 1 cup of mozzarella. Pour a heaping 1 cup of tomato sauce on top, then sprinkle with ¼ cup of the Parmesan cheese.
Add the remaining noodles the same way (4 horizontally and 1 vertically), then layer on the rest of the pesto-ricotta mixture, the remaining roasted veggies, 1 cup of the mozzarella, the remaining tomato sauce, and sprinkle with remaining Parmesan cheese. Finally, top with the remaining 1 ½ cups mozzarella.
Bake the lasagna: Coat a large piece of foil with cooking spray. Place sprayed-side-down on the baking dish and cover tightly (this will prevent the cheese from sticking to the foil). Transfer to the oven on the middle rack and bake for 25 minutes. Remove the foil and dollop the pesto over the top in 1½ teaspoon spoonfuls. Return to the oven and bake uncovered until the cheese starts to get just BARELY golden brown in places, another 15 to 20 minutes.
Garnish with extra Parmesan, if desired, and the basil. Cool for at least 15 to 20 minutes before cutting and serving.
To store: Keep any leftover lasagna slices in an airtight glass container in the fridge for up to 5 days. Reheat in the microwave until heated through.