1. Preheat broiler. In a large frying pan, heat 2 tablespoons of oil over medium-low heat. Add onions and cook, stirring often, until very soft and pale gold, about 10 minutes. Add garlic and cook 1 minute. Remove from pan using a slotted spoon.
2. Season chicken with salt and pepper. Thinly slice crosswise. Add remaining 1 tablespoon oil to pan. Cook chicken over medium-high heat, turning, until white throughout, about 3 minutes. Return onions to pan. Cover and keep warm.
3. Split rolls lengthwise and spread with butter. Broil, cut sides up, about 30 seconds, until golden. Pile chicken and browned onion onto bottoms of rolls. Top with cheese and broil until cheese is melted and bubbly, 30 to 45 seconds. To serve, put tops on sandwiches and cut each sandwich in half crosswise.