Remove ducks' legs and breasts, saving remaining bones and wings for brown duck stock. Leaving bones and skin attached, chop breasts into halves. With a cleaver, remove tips of drumsticks to use in stock.
Sprinkle all over with salt and pepper, and set aside at room temperature 45 minutes. Then place duck pieces in a Dutch oven with rendered fat, garlic and thyme. Cook over low heat, uncovered, 1 1/2 to 2 hours. To test for doneness, pierce with a sharp fork. It should just fall off fork when shaken.
Transfer duck pieces to a medium baking dish and add the fat. Let cool to room temperature, cover with plastic wrap, and refrigerate at least 24 hours or as long as 4 weeks. Before serving, lift duck pieces out of fat. Remove and discard skin and any excess fat by warming slightly on a rack in the oven. Serve duck hot or cold.