Remove ducks' legs and breasts, saving remaining bones and wings for stock. Leaving bones and skin attached, cut breasts into halves.
Carefully trim off as much fat and excess skin as possible and set aside. (There should be quite a bit!) Dice trimmings into small pieces, 1/4 inch or so in diameter.
Sprinkle duck pieces all over with salt and pepper, and set aside at room temperature 45 minutes.
Place fat and skin trimmings in a deep, heavy pan and place over medium heat. Once fat begins to render, increase heat to medium or medium-high. Adjust heat lower if it begins to smoke. Continue rendering until the bits of skin are well-browned, but not burned. Turn off heat and let cool slightly. Strain rendered duck fat and set aside, reserving the crisp bits for other use.
Place duck pieces in a Dutch oven or heavy covered pan 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 completely cover with the fat. Let cool to room temperature, cover with plastic wrap, and refrigerate at least 24 hours or as long as 4 weeks.
To serve, remove duck pieces from fat and warm slightly. Serve duck hot or cold.