Preheat the oven to 425°F (220°C). Line three baking sheets with parchment paper.
Place all grape tomatoes on one prepared sheet, leaving them whole. Spread the sliced mushrooms, broccoli florets, chopped sweet onion, red and orange bell peppers, and chopped zucchini evenly across the other two sheets without crowding.
Drizzle the vegetables on each sheet with the olive oil, season with salt and pepper, and toss to coat. Roast for 25–30 minutes, until vegetables are tender and tomatoes are bursting.
While vegetables roast, combine the ricotta, grated Parmesan, dried basil, dried oregano, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a bowl and mix until smooth.
Combine the roasted non-tomato vegetables in a bowl for easy layering; keep the roasted tomatoes mostly separate, reserving some for the top.
Spray a 9×13-inch baking dish with nonstick spray. Spoon about 1/2 cup marinara sauce on the bottom of the dish and arrange 3 no-boil lasagna noodles in a single layer over the sauce.
Spread one third of the roasted vegetables (except tomatoes) over the noodles. Drop spoonfuls of the ricotta mixture over the vegetables, pull apart one burrata ball and scatter pieces evenly, then sprinkle with a handful of the grated provolone/mozzarella mixture. Spoon a few tablespoons of marinara over, then add 3 more noodles.
Repeat the layering two more times (vegetables, ricotta mixture, torn burrata, grated cheeses, sauce, noodles) so you have three stacked vegetable/cheese layers in total.
Finish with a final top layer of noodles, smother with remaining marinara, sprinkle with the remaining grated provolone and mozzarella, and top with the reserved roasted grape tomatoes.
Place the baking dish on a baking sheet and reduce oven temperature to 400°F (200°C). Bake for 45 minutes, then remove from oven and let the lasagna rest for 45–60 minutes before slicing to allow it to set.