Heat broiler. Roast peppers on a baking sheet, turning occasionally, until blackened, 10 to 15 minutes. Transfer to a bowl and cover with plastic wrap for 10 minutes to steam and loosen skin. Scrape blackened skin from peppers, then seed. Slice into 2-by-1/4-inch strips and set aside.
Meanwhile, reduce oven heat to 350°F. In a small bowl, combine 3 tablespoons olive oil with thyme, oregano, and 1/2 teaspoon each salt and pepper. Brush both sides of eggplant slices with oil mixture and place in single layers on 3 baking sheets. Bake, in batches if necessary, until lightly browned and cooked through, 20 to 25 minutes per batch.
In a medium nonstick skillet over medium heat, cook onion and 4 garlic cloves with remaining oil until golden, about 8 minutes. Add marinara sauce, reduce heat to medium-low, and simmer for 20 minutes.
Meanwhile, in a medium bowl, mix together ricotta, Parmigiano-Reggiano, remaining garlic, and parsley. Season with salt and pepper. Spread about 1 1/4 cups marinara sauce in the bottom of a 9-by-13-inch baking dish before assembling 3 layers. For first layer: Place one-third of eggplant slices over sauce. Dollop one-third of ricotta mixture over eggplant, then sprinkle on one-third of mozzarella. Drizzle with 1-1/4 cups sauce and add half the peppers. Repeat the same process, layering on eggplant, ricotta, mozzarella, sauce, and peppers. For third layer: Add remaining eggplant, ricotta, marinara sauce, and mozzarella.
Bake lasagna until sauce is bubbling and mozzarella is melted and beginning to brown, 40 to 45 minutes.