Set 10-inch cast iron skillet over medium-high heat and melt 2 tablespoons butter. Once melted, add zucchini, summer squash and carrots. Cook until softened, about 5 minutes. Add spinach, a few handfuls at a time, and cook, stirring often until wilted down. Season with 1 teaspoon salt and 1/2 teaspoon black pepper and cook for a minute longer. Transfer to plate or bowl and set aside.
Return skillet to heat and add remaining 3 tablespoons butter. Once melted, add garlic and cook for about 30 seconds. Stir in tomatoes and cook until softened, about 5 minutes. Season with remaining salt, pepper, half the basil and half the parsley. Simmer on low for about 2 minutes longer. Remove from heat and carefully transfer to blender and purée until smooth.
In medium bowl, stir together ricotta, egg, parmesan and remaining herbs until evenly combined. Season with a pinch of salt and pepper.
To assemble lasagna, place about 1 cup of sauce onto bottom of same skillet you used earlier, no need to clean. Top with 2 pieces of lasagna noodle, you might have to break them a bit to have them fit. Take another lasagna sheet and break it into small pieces and place over any holes on bottom. It doesn’t have to look perfect, but you want it fairly well covered. Top with half of veggies, spread with half of ricotta mixture and sprinkle with a bit of shredded mozzarella.
Place two more sheets on top, and cover up any holes again with broken pasta pieces. Spread sauce over top and repeat layering with remaining veggies, ricotta, a bit more cheese and two more lasagna sheets. Finish off with sauce on top and shredded mozzarella cheese.
Set skillet over medium-low heat, cover and simmer until noodles have softened and cheese has melted, about 20 minutes. Before serving, you can place lasagna under broiler for a minute or two to brown the top, but it’s not necessary. Let sit for about 10 minutes before cutting into it, allowing the sauce to thicken slightly.