1. Rinse zucchini; trim off and discard ends. Cut zucchini into 1/2-inch slices. Trim off and discard stem ends of mushrooms. Rinse mushrooms, drain, and cut into quarters. Cut onion crosswise into 1/4-inch slices.
2. In a 5- to 6-quart nonstick pan over high heat, frequently stir zucchini, mushrooms, onion, garlic, and 1/2 cup broth until broth has evaporated and vegetables are lightly browned, 12 to 15 minutes. Add 1/2 cup more broth and stir to release browned bits. Stir in tomatoes and pour vegetables into a bowl; keep warm.
3. To pan, add remaining broth, polenta, and corn. Stir over high heat until boiling, 3 to 4 minutes.
4. Spoon polenta into wide bowls. Top equally with vegetable mixture. Sprinkle cheese and green onions over portions.