Cover the duck with salt and pepper on both sides. Place two of the legs on a plate with skin side down. Cover with several sprigs of thyme and cloves of garlic, and then with the other two legs, skin side up. Sprinkle several more thyme sprigs and the remaining garlic over the duck, cover with plastic wrap, and refrigerate overnight to cure.
Move the duck, thyme, and garlic to a heavy pot, just wide enough to hold the four legs in a single layer. Melt the fat and pour it into the pot. (The duck should be completely submerged.) Cook in a 250 ° F. oven for about five hours, until the meat is easily removed from the bone with a fork. Discard the bones. Return the fat to its container and refrigerate. Put the meat into a small bowl with the garlic. Cover and refrigerate until thoroughly chilled, for up to several days.
Cut the chilled meat into small pieces and place in a large mixing bowl with the garlic, mustard, parsley, onion, peppercorns, and cognac. Remove the fat from its container and separate from the layer of gelatin at the bottom. Add a half cup of fat to the bowl and stir with a wooden spoon as the meat turns to a paste, adding some of the gelatin (being careful not to add so much as to oversalt the dish) and more fat as needed to produce a creamy texture.
Put the rillettes in a crock (or several crocks), leaving 1/4 inch of space at the top. Cover with a few sprigs of thyme. Melt 1/2 cup of the fat and pour over the rillettes. Refrigerate until thirty minutes before serving time.