Cut the duck in quarters and remove the drumstick tips. Season with 1 teaspoon each of the salt and pepper, rub all over with the garlic and place a piece of cinnamon stick on each quarter. Let sit at room temperature 45 minutes.

Preheat the oven to 350 degrees.

Transfer the duck to an ovenproof frying pan along with the chicken or other fat. Bring to a boil over medium-high heat, cover with foil and transfer to the oven. Bake 1 hour and 15 minutes. Set aside to cool.

When cool enough to handle, lift the duck from the fat and remove and discard the skin. Cover with a wet towel until serving time. To reserve, store the duck in the cooking fat and refrigerate up to a week.

To reheat the duck, remove from the fat if necessary, and place on a rack in a tightly covered pan. Warm in a 325 degree oven for about 15 minutes.

Meanwhile, to make the lime onions, heat the olive oil in a medium skillet over high heat. Saute the onions, stirring frequently, until just wilted (but not browned). Toss with the lime juice and the remaining 1 teaspoon salt and l/2 teaspoon pepper. Keep warm. The onions can be made a day in advance and refrigerated. Reheat before serving.

To serve, spread a bed of lentils on each serving plate. Top with a piece of duck and smother with the warm onions. Serve immediately.