1. Preheat oven to 375 degrees.

2. Sauce: Heat oil in a large saucepan over medium-high. Add mushrooms and eggplant and saute 10 minutes. Add onion, carrots, celery, and 1 teaspoon salt; cook until vegetables have browned in places and most of moisture is cooked out of mushrooms and eggplant, about 15 minutes. Add garlic, oregano, and red-pepper flakes; cook until fragrant, 1 minute. Add tomato paste; cook 2 minutes more. Puree tomatoes in a blender or food processor; add to pan. Bring to a boil, then reduce heat to low and simmer, stirring occasionally, until reduced and thickened slightly but still very loose, about 15 minutes. Remove from heat.

3. Cheese Mixture: In a large bowl, whisk together ricotta, milk, eggs, 1 cup Parmigiano-Reggiano, 2 1/2 cups mozzarella, parsley, 1 teaspoon salt, and 1/4 teaspoon pepper.

4. Assembly: Spread 2 cups sauce evenly in a 9-by-13-inch baking dish, preferably ceramic or enameled cast iron. Arrange 3 lasagna noodles over sauce (space noodles evenly; they will expand as the lasagna cooks). Spread 2 cups cheese mixture over noodles; top with 3 more noodles. Repeat layering of vegetable and cheese mixtures between noodles twice more. Spread 2 cups sauce evenly over top layer of noodles. Sprinkle evenly with remaining 1 1/2 cups mozzarella and 1/4 cup Parmigiano-Reggiano. Loosely cover with parchment-lined foil and transfer to a rimmed baking sheet.

5. Bake until noodles are tender but still al dente (easily pierced with the tip of a knife), 40 to 45 minutes. Uncover; continue baking until bubbly and puffed slightly along edges and golden brown in places on top, 20 to 25 minutes more. (There may appear to be a lot of liquid in the dish. It will be absorbed as the lasagna cools.) Transfer dish to a wire rack and let cool slightly and set, about 20 minutes; cut into squares to serve.