In a shallow bowl, whisk soy sauce, brown sugar, vinegar, oyster sauce, garlic, ginger, and cornstarch-water mixture.
Fold mushrooms and sweet peppers into marinade until evenly coated and season with pepper to taste.
Cover bowl with plastic wrap and transfer to refrigerator. Allow mushrooms and peppers to marinate at least 2 hours or up to 4, tossing occasionally to evenly coat in marinade.
When ready to grill, use a slotted spoon to remove mushrooms and peppers from marinade (reserve 2 tablespoons marinade). Preheat an outdoor grill to 400F OR prepare a large nonstick pan to use indoors on stovetop by greasing well.
Grill mushrooms and peppers on a sheet of nonstick aluminum foil or in a grill pan on outdoor grill OR on stovetop 6-8 minutes, tossing occasionally until mushrooms and peppers are slightly charred.
Transfer warm mushrooms and peppers to a serving bowl, drizzle with reserved marinade, and toss until evenly coated. Top with sliced green onions. Serve warm and enjoy!