Preheat the grill to medium-high heat. Heat 1 Tbsp of the oil in a non-stick pan over medium heat. Add the mushrooms and garlic and cook until the mushroom are tender, about 4-5 minutes. Brush the chicken with the remaining with oil; season with oregano, paprika, salt and pepper. Grill 3-4 minutes per side, or until cooked through. When cooked, top each breast with 1/4 cup of cheese and 1/4 of the mushrooms. If cooking on the barbecue, close the lid and cook until the cheese is melted. If cooking on a stovetop grill pan, place under your ovens broiler until the cheese is melted. Spread the mayonnaise on the buns. Top bottom halves of the buns with the salad greens. Set on the chicken and the top buns and serve.