How to Make It
In a bowl, toss the chicken thighs with 2 tablespoons of the olive oil, the rosemary, garlic and a generous pinch of salt. Cover and let the chicken stand at room temperature for 30 minutes or refrigerate for 4 hours or overnight.
Meanwhile, in a large, deep skillet, melt the butter in the remaining 1 tablespoon of olive oil. Add the sliced onions, thyme, vinegar, sugar and a generous pinch of pepper. Cook over moderately low heat, stirring occasionally, until the onions are softened, about 30 minutes; add a few tablespoons of water from time to time to keep the onions moist.
Light a grill and preheat the oven to 500°. Season the chicken with salt and pepper and grill over moderately high heat, turning, until lightly charred and cooked through, about 15 minutes. Let the chicken cool, then pull the meat from the bones and tear into shreds.
Arrange 6 tortillas on 2 large baking sheets. Spoon the onions onto the tortillas and top with the shredded chicken and blue cheese. Cover with the remaining tortillas, pressing to flatten slightly. Bake for about 6 minutes, until the quesadillas are golden and toasted. Transfer the quesadillas to plates and serve right away.