1. Cut meat into 3/4-inch chunks.
2. Trim and discard eggplant stems. Cut eggplant into 3/4-inch chunks.
3. Stem and seed bell pepper. Cut into 3/4-inch chunks.
4. Peel onion and cut into 3/4-inch chunks.
5. Place a 5- to 6-quart pan over high heat. Add 2 teaspoons oil and swirl to coat pan. Add pork and garlic; stir occasionally until lightly browned, 10 to 15 minutes. Pour from pan into a bowl.
6. To pan, add 1 teaspoon oil, onion, carrots, and mushrooms; stir-fry over high heat until onion begins to brown, about 10 minutes.
7. Add hoisin sauce, wine, broth, and pork. Stir until boiling to release browned bits in pan. Cover and simmer over low heat 10 minutes; stir often.
8. Add eggplant and bell pepper to pan. Mix, cover, and simmer over low heat until pork is very tender when pierced, 30 to 40 minutes longer; stir occasionally.
9. Add sesame oil and soy sauce to taste. Garnish portions with green onions.