Preheat the oven to 425°F (220°C) and line three baking sheets with parchment paper.
Place the grape tomatoes on one sheet (leave whole). Spread the mushrooms, broccoli, chopped onion, red and orange peppers, and zucchini evenly across the other two sheets without crowding.
Drizzle the vegetables and tomatoes with the olive oil, season with salt and pepper, and toss each pan to coat.
Roast all pans for 25 to 30 minutes, until the tomatoes are bursting and the other vegetables are tender. Remove from oven.
In a bowl, stir together the ricotta, grated Parmesan, dried basil, dried oregano, 1/4 teaspoon salt, and 1/4 teaspoon pepper until combined.
Reduce oven temperature to 400°F (200°C). Spray a 9×13-inch baking dish with nonstick spray and spread about 1/2 cup marinara on the bottom.
Arrange 3 no-boil lasagna noodles over the sauce to start the first layer.
Spread one third of the roasted mixed vegetables (exclude the whole roasted tomatoes) over the noodles, dollop one third of the ricotta mixture over the vegetables, pull apart about one burrata ball and scatter pieces evenly, then sprinkle a portion of the grated provolone and mozzarella.
Spoon a few tablespoons of marinara over the cheeses, then place 3 more noodles and repeat the layering two more times so you have three vegetable/ricotta/burrata layers in total.
Finish with a top layer of lasagna noodles, smother with remaining marinara, and sprinkle with the remaining grated provolone and mozzarella. Scatter the reserved roasted grape tomatoes on top.
Place the lasagna dish on a baking sheet and bake at 400°F (200°C) for 45 minutes.
Remove from oven and let rest for 45 to 60 minutes before slicing and serving.