Place half of eggplant in a colander, sprinkle lightly with salt, add remaining eggplant and sprinkle with salt. Let colander stand in sink for 10 minutes to remove eggplant's bitterness. (The salt will be rinsed off later.)
Bring a large pot of salted water to a boil, add spaghetti and cook according to package directions, stirring often, until al dente. While water is coming to a boil, begin preparation of sauce: Bring 1 cup water to a boil in a large skillet. Add green beans and broccoli, cover and cook over medium heat until tender, about 6 minutes. Drain vegetables and rinse with cold water.
Rinse eggplant with cold water and pat dry on clean kitchen towel. Add 3 Tbsp. oil to skillet and place over mediumhigh heat. Add eggplant and cook, stirring, until golden, about 3 minutes. Add remaining 1 Tbsp. oil, zucchini, bell pepper and garlic; cook, stirring, for 3 minutes. Add tomato sauce and water, then simmer, stirring occasionally, until sauce is thickened and vegetables are tender, about 5 minutes. Add green beans and broccoli and heat through.
Drain spaghetti and transfer to a large serving bowl. Add vegetable sauce, freshly ground pepper and Parmesan; toss well.