Cook the lasagna noodles in salted water according to the package instructions. Drain, reserving 1/3 cup of pasta water for the sauce.
In a medium bowl, mix together the ricotta, parmesan, and 2 cups of the mozzarella cheese. Stir until mixed and set aside.
Preheat oven to 425°F.
Chop the vegetables, keeping the carrots separate. Add the olive oil to the skillet and turn the heat to medium-low. Add the carrots and cook for a couple of minutes. Then add the onions, broccoli, mushrooms, and zucchini. Cook until tender, about 5 minutes. Add the garlic and cook until fragrant, about 30 seconds to 1 minute. Do not brown the garlic.
If the lasagna noodles are too hot to handle, rinse them in cool water and drain.
Spray the casserole dish with cooking spray. Add a small amount of pasta sauce (about 1/4 cup) to the bottom of the casserole dish.
Stretch out the lasagna noodles on a cutting board or countertop. Divide the cheese mixture between the noodles and spread it evenly on each one.
Divide the cooked vegetables between the noodles and add them evenly on top of the cheese.
Add a little pasta sauce to each noodle. Then start with one end, carefully roll the noodle, and place it in the baking dish. Repeat for all the noodles.
Spoon the rest of the pasta sauce over the noodles. Then add 1/3 cup of the pasta water around the noodles, being careful not to wash the sauce off. Then sprinkle the remaining 1 cup of mozzarella cheese over the noodles.
Bake for 20-30 minutes, until the roll-ups are hot and the cheese on top is bubbly and beginning to brown. If the cheese starts to get too brown, place a piece of aluminum foil over the baking dish.