Preheat oven to 375 degrees. Spray a 9 x 13 baking dish with cooking spray, set aside.
Heat olive oil in a large skillet or dutch oven over medium high heat. Add onion and cook, stirring occasionally, until softened, about 3 minutes. Add in garlic, zucchini, mushrooms, and kosher salt. Cook until slightly softened, about 7 minutes.
Stir in roasted red bell peppers, tomato sauce, and diced tomatoes. Bring to a low simmer and cook for 8-10 minutes, until liquid has slightly reduced. Add in chopped basil. Using an immersion blender, puree the sauce until desired consistency is reached. Set aside.
Make the ricotta filling: stir ricotta (or cottage cheese), egg, and ¼ cup of parmesan cheese in a bowl, set aside.
Make the shredded cheese: combine the mozzarella and remaining 1 cup of parmesan in a bowl, set aside.
To prepare the lasagne, spoon about a cup of the sauce into the baking dish to lightly cover the bottom. Arrange 4 noodles on top of the sauce. Spread on ⅓rd of the ricotta filling, ¼th of the shredded cheese, then 2 cups of the sauce. Repeat layers three times. For the final layer, add sauce on the noodles and sprinkle with remaining shredded cheese.
Cover with foil and bake for 30 minutes. Remove foil then bake 15 minutes until cheese is bubbly. To brown the cheese slightly, slide under a broiler for 1 to 2 minutes. Let lasagne rest 10 to 15 minutes before serving.
Notes
Make-ahead vegetable lasagne and refrigerate unbaked up to 2 days in advance. Sauce can also be made up to 3 days in advance. Alternatively, you can freeze in an airtight container for up to 3 months.