Simmer vinegar, sugar, 1/2 cup water, and the oregano in a small saucepan, stirring occasionally, about 2 minutes. Add onions and simmer, 2 minutes. Transfer mixture to a bowl and chill, loosely covered, about 1 hour.

Preheat oven to 400 degree F. Toss chicken with salsa until combined well. Divide and layer half of cheese, the chicken mixture, then the remaining cheese on bottoms of buns and arrange on a shallow baking sheet with bun tops on side, cut sides up. Roast in middle of oven until cheese melts and buns are golden brown, about 5 minutes. Remove from oven and divide strained onion mixture over melted cheese; cover with bun tops. Serve remaining salsa on side. Makes 4 servings.