1. Season chicken with marjoram and 1/4 teaspoon each salt and pepper. In a large frying pan, cook chicken in 2 tablespoons of olive oil over medium heat until tender and white throughout, about 5 minutes. Remove chicken from pan. Heat remaining 1/4 cup oil in same pan. Add onion, eggplant, zucchini, bell pepper, and garlic and cook, stirring occasionally, until onion is softened, about 5 minutes. Cover pan and simmer 20 minutes. Add tomatoes, capers, and basil and simmer 10 minutes, covered. (If sauce seems too juicy, uncover during last 5 minutes of cooking to reduce) Return chicken to sauce to reheat.
2. Meanwhile, cook pasta in a large pot of boiling salted water until tender but still firm, about 10 minutes. (If using precooked lasagna noodles, reconstitute according to package directions) Carefully drain pasta in a colander, being careful that the noodles do not tear.
3. Preheat broiler. In an oiled 9x13-inch baking dish or in 6 individual oven-proof gratin dishes, make a layer of one-third of noodles, cutting to fit if using individual dishes. Add a layer of half of chicken and half of sauce. Make another layer using half of remaining lasagna noodles and all of remaining chicken and sauce. Cover with remaining lasagna noodles. Sprinkle with cheese over top.
4. Broil about 4 inches from heat, until cheese is melted and lightly browned, about 1 minute.