Preheat the oven to 350°F. Heat 1 tablespoon of the oil in a large skillet over medium-high heat.

Add eggplant and 1/4 teaspoon of the salt, and cook, stirring occasionally, until eggplant is browned, about 7 minutes.

Transfer to a large bowl.

In the same pan, add the remaining 1 tablespoon oil.

Add mushrooms and 1/4 teaspoon of the salt, and cook until browned, about 5 minutes.

Transfer to the bowl with the eggplant.

In the same pan, cook onion and peppers until softened, about 7 minutes.

Add squash, garlic, and 1/4 teaspoon of the salt, and cook until squash just begins to soften, about 4 minutes.

Add vinegar and bring to a simmer; cook until vinegar is almost nearly reduced, about 4 minutes.

Add tomatoes and oregano, stir to combine, then remove from the heat.

Combine with reserved eggplant and mushrooms, and set aside.

In a small bowl, combine ricotta, egg, and black pepper; stir to combine.

In a separate bowl, combine mozzarella and parmesan cheeses.

Spread 2 heaping cups of the vegetable mixture in the bottom of a 9x15-inch baking dish.

Cover with a layer of lasagna noodles.

Spread 1/3 of the ricotta mixture (about 1/2 cup) over the noodles, then sprinkle with 1/3 of the cheese mixture.

Repeat with the remaining noodles, ricotta, and cheeses, ending with a layer of cheese.

Set aside any remaining vegetable mixture to top the cooked lasagna later, if you like.

Cover the lasagna with foil and cook until cheese is bubbly and hot throughout, about 30 minutes.

Remove foil and bake until the top is lightly browned, about 5 minutes.

Set aside to rest for 10 minutes before slicing, and serve with reserved vegetable mixture, if you like.