Cook the duck, skin-side down, covered, in a large heavy nonstick skillet over moderately-low heat until the skin is well-browned and crisp and the duck is heated through, 15 to 20 minutes. Remove the duck and pour off all but 1 teaspoon of the fat from the pan and reserve. Cook the onions in the remaining duck fat over moderate heat until crispy and brown, about 10 minutes. Add 3 tablespoons water to the pan and scrape the bottom of the pan until the water evaporates. Remove the skin from the duck and reserve. Shred the meat and place in a medium bowl.
Place the duck skin back in the pan and cook over a moderate heat, until the skin is crispy, about 5 minutes. Remove the skin from pan, chop and put in the same bowl as the shredded meat. Pour off all the duck fat from the pan and reserve the fat.
Place 1 tortilla on your work surface and spread 1 tablespoon goat cheese, 2 tablespoons of the onions, 1/2 cup of the duck mixture and 1 tablespoon scallions on top of a tortilla. Lay another tortilla on top of the scallions.
Brush a nonstick skillet with 1/2 teaspoon of the reserved duck fat over moderately high heat until hot but not smoking. Cook the quesadilla turning once, until golden, about 6 minutes. Repeat with the remaining ingredients and serve immediately.
BYOC: What can't you throw in a quesadilla? Try some Monterey Jack cheese or some pickled jalapenos.