(5- to 5 1/2 pound) Pekin duck (sometimes called Long Island duck), including neck
tablespoons Armagnac or Cognac
cups plus 2 tablespoons water
medium shallots, sliced
garlic cloves, halved
carrots, 1 cut into 1/4-inch-thick slices and 1 finely chopped
sprigs fresh thyme
large bay leaf (not California)
3/4 teaspoons kosher salt
/2 teaspoon whole black peppercorns
1/2 teaspoons unflavored gelatin
tablespoons finely chopped fresh flat-leaf parsley
Accompaniment: 1/3-inch-thick baguette slices, toasted
Cut breasts from duck and discard skin and all visible fat. Reserve legs and carcass for another use.
Brown duck neck (halved if necessary) in a dry 2- to 3-quart heavy saucepan over moderate heat, turning once, about 10 minutes total. Transfer to a plate. Pat duck breasts dry and season with salt and pepper, then add to pan and brown over moderate heat, turning once, about 8 minutes total. Transfer to plate with duck neck.
Add Armagnac to pan and deglaze by boiling, scraping up any brown bits, until most of liquid is evaporated. Add 2 cups water, shallots, garlic, sliced carrot, thyme, bay leaf, kosher salt, peppercorns, and duck breasts and neck with any juices on plate and cook at a bare simmer, covered, 1 hour. Discard neck, then transfer breasts to a plate and cool. Shred duck meat, then chop.
Pour broth through a sieve into a bowl, discarding solids. (If liquid measures less than 2 cups, add water.) Return to cleaned pan and add chopped carrot. Simmer, covered, until carrot is tender, 3 to 5 minutes.
While carrot is cooking, sprinkle gelatin over remaining 2 tablespoons water in a cup and let soften 1 minute.
Stir softened gelatin into hot broth until dissolved, then stir in duck meat, parsley, and salt and pepper to taste. Ladle into crocks (you’ll have about 3 cups total) and chill, covered, until set, 3 to 4 hours. Let stand at room temperature 20 minutes before serving.