Handle the chicken as little as possible, optionally wearing gloves, to prevent cross contamination. Dress the skin on each to cover the flesh. Leave any fat deposits except exceptionally large ones. Place the thighs on a thick layer of paper towels on a cutting board, skin side up. Cover the thighs with another thick layer of paper towels and press onto the surface of the skin. The chicken should be as dry as possible, for best searing without sticking.

Uncover the chicken after no less than two or three minutes and discard the paper towel that was on top. Wash your hands thoroughly with soap and warm water.

Generously cover the skin first with the sea salt, and then with a generous grind of black pepper, to your preferred level of coarseness. I prefer a coarser grind.

Allow the chicken to sit while you prepare the cooking pot.

Heat the pot over medium high, large enough to hold all the chicken comfortably in the bottom in a single layer.

As the pot begins to heat up, add the duck fat and the olive oil to the bottom, and swirl occasionally to coat the bottom. When the fat is ready it will shimmer and will be on the verge of smoking.

Carefully place each thigh in the hot fat, using tongs if you need them, salted and peppered skin side down. The chicken will begin to sputter immediately. Work quickly and when all the thighs are in the pot, adjust their positions to make sure they each have some room. You can partially cover the pot to help minimize the fat splatter while it cooks. Before covering, though, salt and pepper the exposed fleshy side of the thighs evenly.

Check the thighs occasionally. They should eventually brown to a deep golden color. This will take from six to eight minutes. Halfway through the estimated time, turn each thigh 180° to ensure even browning. After the skin side is browned satisfactorily, turn each thigh over and repeat the process of browning, though the reverse side will take less time. Turn each thigh through 180° on this step also.

When the thighs are thoroughly browned on both sides, remove them, setting them aside on a plate or platter in a single layer. And remove the pot from the heat. Extinguish the burner to prevent accidents.

Carefully remove the hot liquid fat from the casserole, and reserve three tablespoons of it, which will be returned to the pot. Discard the remaining fat safely or reserve it for some other use (it is a combination of the original duck fat or butter, EVOO, and the rendered chicken fat, and is a particularly rich fat for browning foods for other dishes, where appropriate).

Replace the pot on the burner and turn up to medium-high. Add the reserved 3 tablespoons of cooking fat. It should take only a minute or two to reach temperature. But watch it carefully and be sure not to let it begin to smoke. Add the split and halved shallots carefully distributed so they can all brown at once. Move the shallots around with tongs and as each section browns turn it to brown another side of it.

Browning all the shallots slightly should take only a minute or two.

Add the artichoke hearts, evenly distributed in the fat in the pot, and with the tongs make sure each is coated in fat. Keep the shallots and artichoke hearts moving in the pot. The artichoke hearts will begin to brown immediately. Give them only a minute to brown.

Add the white wine and stir. Using a wooden spoon, as the wine boils, stir the ingredients, scraping the bottom of the pan. All browned bits should dissolve and come off easily. Stir until the wine is almost all boiled off and lower the heat to medium.

Add the chicken stock and stir gently with the wooden spoon. As the liquid begins to simmer, add, in any order, the balsamic vinegar, the thyme, the basil, the two kinds of mustard, and keep stirring as they dissolve and distribute.

When the mixture has reached a steady simmer, lower the heat to medium low.

Now, carefully and gently place the four chicken thighs in the casserole evenly, skin side up. Be sure to add the liquid that has collected in the plate on which you reserved the chicken to the pot and stir it in. The chicken thighs should sit in the liquid comfortably without being submerged. The rest of the ingredients should rise no higher than half-way up their sides.

Make sure the simmer continues gently, and cover the pot.

After about half the remaining time has elapsed, add the tomatoes to the pot, distributing them around and between the chicken. Gently use the wooden spoon or tongs to coat the tomatoes with the liquid in the pot. Re-cover, and allow to continue to simmer.

When the chicken is completely cooked, which should take no more than a half-hour, remove the thighs to a heated platter or plate. Unless the thighs you used are unusually large, or you did not preserve the simmer on the covered casserole, the meat should be cooked through and tender. To be absolutely certain, use an instant read meat thermometer, with a probe, to test the temperature, being careful to test the thickest part of the thigh and not to touch the bone with the probe. Chicken is done at 165°F.

Once removed from the pot, the chicken will continue to cook to even more tenderness and doneness and the juices will recede.

Meanwhile raise the temperature under the pot to medium or medium high, so that it simmers more forcibly and allow it to cook further, in order to reduce and thicken. Remove it from the heat after five minutes.

You may replace the chicken into the pot, or serve this dish separately from the two containers: the chicken from the platter, and the stew from the pot.

This dish is excellent served with broad egg noodles, or with whole grains (like barley, farro, or wheat berries), or with whole grain (i.e., brown) rice, either short of long grain.

Serve each diner in a deep-bowled plate, first with a portion of the grain or pasta, place a chicken thigh on top of the grain, and then ladle or spoon a portion of the other ingredients over the chicken and grain.