Preheat oven to 500 degrees. Sprinkle 2 tablespoons salt over eggplant slices, and let drain for 45 minutes.
Meanwhile, place zucchini in a single layer on oiled baking sheets. (Work in batches if necessary.) Brush with oil, and season with salt and pepper. Bake, rotating pans halfway through, until tender and just starting to brown, 10 to 12 minutes. Let cool slightly, then transfer to a dish.
Place onion rings in a single layer on oiled baking sheets. (Work in batches if necessary.) Brush with oil, and season with salt and pepper. Bake until tender and browned, 10 to 15 minutes. Let cool slightly, then chop into 1/2-inch pieces. Mix with goat cheese in a small bowl.
Gently press eggplant slices between double layers of paper towels to remove excess liquid. Place eggplant in a single layer on oiled baking sheets (work in batches if necessary.) Brush with oil, and season with pepper. Bake until tender and golden, 6 to 9 minutes.
Reduce oven temperature to 375 degrees. Spread 1/2 cup of marinara in a 9-by-13-inch baking dish. Place a layer of noodles over marinara. Layer eggplant on top of noodles. Spread 1/2 of the goat cheese mixture over eggplant. Spread 1 cup marinara over cheese layer, and sprinkle with 2 tablespoons marjoram. Repeat layering once, replacing eggplant with zucchini (noodles, zucchini, goat cheese, marinara, marjoram). Place a layer of noodles on top. Spread remaining marinara on top of noodles, and sprinkle with Pecorino Romano cheese.
Place baking dish on a rimmed baking sheet. Bake until cheese is golden and sauce is bubbling, about 45 minutes. Let stand for 15 minutes before slicing and serving.