Preheat the oven to 375°F (190°C).
Place a 12-inch (or larger) oven-proof skillet over medium heat and add 2 tablespoons olive oil.
Add the diced onion and cook, stirring frequently, until it begins to soften, about 3–4 minutes.
Add the diced garlic and sauté for 1 minute until fragrant.
Add the diced zucchini and sliced mushrooms and cook 3–4 minutes until zucchini is tender and mushrooms soften.
Pour 3 cups Italian tomato sauce over the vegetables, stir to combine, and cook until warmed through.
Stir in the baby spinach and cook about 1 minute until wilted; season with salt and pepper to taste.
Remove the skillet from heat and carefully scoop out all but 1 cup of the sauce, leaving the remaining sauce evenly spread on the bottom of the skillet.
Cover the sauce with a single layer of broken lasagna noodle pieces (no need to be perfect), covering most of the sauce.
Drop 5–6 mounds of ricotta (about one-third of the 15-ounce container) over the noodles, then spread one-third of the remaining sauce over the ricotta and sprinkle one-third of the shredded mozzarella.
Repeat layering: noodles, one-third ricotta, one-third sauce, and one-third mozzarella, then add a final layer of noodles and the remaining sauce.
Top with the remaining ricotta, the rest of the shredded mozzarella, and sprinkle 1/4 cup shredded Parmesan evenly on top.
Transfer the skillet to the preheated oven and bake about 30 minutes, until cheese is bubbly and pasta is cooked through.
Remove from oven and let cool about 5 minutes before serving.