Preheat the oven to 450ºF.
Arrange the eggplant and zucchini slabs in a single layer on one large baking sheet and the bell pepper slabs and unpeeled garlic cloves on another. Sprinkle with the oil, 1/2 teaspoon of the salt and 1/4 teaspoon of the black pepper. Rub with your hands to coat the vegetables evenly (yes, your hands will get greasy; just wash them afterwards).
Roast for 20 minutes. Add the mushrooms and roast until the vegetables are browned, soft, and shrunken, 15 to 20 minutes more. Remove from the oven and let cool slightly on the sheets. Cut each slab of eggplant, zucchini, and bell pepper in half lengthwise into strips.
Meanwhile, cook the lasagna noodles in a large pot of lightly salted water until the noodles are slightly chewy, 10 to 12 minutes. Drain, run under cold water, then separate and cool the noodles in a large bowl of ice water. Set aside.
Heat sauce in a medium saucepan over low heat. When cool enough to handle, squeeze the roasted garlic from its skins into the sauce and stir until incorporated.
Reduce the oven to 350°F. In a large bowl, combine the ricotta, 1 cup of the mozzarella, 1/4 cup of the Parmesan, the egg, parsley, thyme, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon black pepper.
Spoon a thin layer of sauce over the bottom of a 13-inch-by-9-inch baking dish. Arrange a layer of noodles in the pan over the sauce. Spread 1/3 of the cheese mixture over the noodles. Spoon 1/4 of the sauce over the cheese. Layer on 1/3 of the strips of roasted eggplant, zucchini, bell pepper, and mushrooms. Repeat the layers of noodles, cheese, sauce, and roasted vegetables to make a total of 4 layers of noodles. Top the final layer of noodles with the remaining sauce, 1 cup mozzarella, and 1/4 cup Parmesan.
Bake until well-browned on top, 30 to 35 minutes. Let stand for 10 minutes before cutting. Serve with thick slices of Panera Bread Tomato Basil Bread.
Tip: To skip a step, use no-boil lasagna noodles and increase the amount of sauce to 4 cups (32 ounces). Be sure to completely douse the no-boil noodles with sauce and cheese so they can soften properly. Switch the veggies to suit your pleasure and be sure to freeze any leftovers for quick meals on busy nights.