Trim the steak of any fat, and slice thinly into strips.
In a nonreactive bowl, combine the soy sauce, 1 Tbsp of the oil, garlic, white wine and 1 tsp of the sugar. Add the beef, and stir to coat the meat. Refrigerate for 30 minutes, or up to eight hours.
In a 10- or 12-inch skillet, heat the oil over high heat, then add the peppers, onion and mushrooms. Cook over high heat, tossing the vegetables, adding the salt, pepper and the remaining sugar, stirring constantly. Continue to cook for about five minutes, or until the vegetables are crisp but tender. Season the mixture with oregano, then set aside.
Drain the steak from the marinade. Heat a wok on high, then add the steak, stir-frying until it loses its pink color. Remove from the heat. Cut the pitas in half, and open each pocke Tbsp Place some of the steak into each pocket, cover with some of the Monterey jack cheese, if used, and then top with the pepper and mushroom mixture.Servings: 2