Add 2 tablespoons of olive oil to a skillet on medium-high heat. Once warmed, add the onion, bell peppers, 4 minced garlic cloves, salt, and pepper. Cook, stirring occasionally, until golden brown and caramelized.
Add the eggplant, yellow squash, zucchini, tomato paste, dried oregano, and 1 tablespoon of thyme, and more salt and pepper. Cook until the vegetables are soft and the tomato paste deepens in color and starts to stick to the bottom.
Add the diced tomatoes and bring the mixture to a simmer. Cook on low for at least 30 minutes. Set aside to prepare the rest of the lasagna layers.
In a large bowl, mix together the ricotta, mozzarella, basil, parsley, salt, pepper and egg. Set aside.
To a bowl, add the sliced Japanese eggplant, yellow squash, zucchini, Roma tomatoes, garlic, 1 tablespoon thyme, salt, pepper, and 2 tablespoons olive oil, and gently toss to combine.
In a 9x13-inch (23x33 cm) glass baking pan, add a layer of the ratatouille to the bottom. Top with lasagna noodles, and spread a layer of the ricotta mixture, then shingle alternating slices of the marinated vegetables. Repeat with another layer of ratatouille, noodles, ricotta, and on the top shingle the vegetables.
Bake for 30 minutes, or until the cheese and sauce are bubbling slightly and the sliced vegetables on top have browned to your liking.