1. Warm oil in a large heavy skillet over low heat. Add onions, salt, pepper and rosemary. Stir to coat. Cover and cook, stirring occasionally, until onions are very soft, 20 to 25 minutes. Uncover skillet, sprinkle sugar over onions and stir to combine. Raise heat to medium and cook, uncovered, stirring often, until onions are an even golden color, 12 to 15 minutes. (Add a bit of water if skillet becomes too dry.)
2. Preheat grill to medium-low. Lay tortillas on a flat surface and spread onion mixture evenly over lower half of each. Sprinkle cheese over onions and fold top half of each tortilla down over filling, pressing gently to adhere.
3. Oil grill. Place quesadillas on grill and cook until lightly browned and cheese has melted, 2 to 4 minutes per side. Transfer to a cutting board and let stand for 5 minutes. Cut each quesadilla into 3 wedges. Serve warm or at room temperature.