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1
Cut breasts from duck and discard skin and all visible fat.
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2
Reserve legs and carcass for another use.
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3
Brown duck neck (halved if necessary) in a dry 2- to 3-quart heavy saucepan over moderate heat, turning once, about 10 minutes total.
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4
Transfer to a plate.
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5
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.
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6
Transfer to plate with duck neck.
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7
Add Armagnac to pan and deglaze by boiling, scraping up any brown bits, until most of liquid is evaporated.
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8
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.
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9
Discard neck, then transfer breasts to a plate and cool.
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10
Shred duck meat, then chop.
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11
Pour broth through a sieve into a bowl, discarding solids.
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12
(If liquid measures less than 2 cups, add water.)
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13
Return to cleaned pan and add chopped carrot.
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14
Simmer, covered, until carrot is tender, 3 to 5 minutes.
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15
While carrot is cooking, sprinkle gelatin over remaining 2 tablespoons water in a cup and let soften 1 minute.
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16
Stir softened gelatin into hot broth until dissolved, then stir in duck meat, parsley, and salt and pepper to taste.
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17
Ladle into crocks (you'll have about 3 cups total) and chill, covered, until set, 3 to 4 hours.
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18
Let stand at room temperature 20 minutes before serving.
