Preheat oven to 350°F/175°C. 

Season chicken and cabbage generously on both sides with salt and pepper. Heat the oil in a heavy 10- or 12-inch lidded skillet over medium-high heat. Add chicken, skin side down. Brown, not moving chicken, until skin is golden and the fat is rendered, 4-5 minutes. Flip chicken and nestle cabbage wedges in between them. Brown until the bottom of the cabbage is browned, 5 minutes, In a liquid measuring cup or bowl combine the water, stock, and the 2 tsp salt & ½ teaspoon pepper. Carefully flip the cabbage, then add the wine, bring to a boil and boil until the wine reduces, 2 minutes. Add the broth/water mixture. Spoon the barley into skillet where the liquid us pooled, arrange the shallots and garlic cloves in the pan, scatter the thyme across the top, and return the skillet to a boil. Cover with parchment, then foil and a lid if you have one. Transfer to the oven and cook until the barley is cooked through (but still al dente), 45 minutes. Remove from the oven, raise the heat to 450° (220° C), uncover, add 1/2 - 1 cup more water if the skillet seems a bit dry, and return to the oven. Cook until the tops is golden and bubbly and the barley is swollen and starchy, 7 to 8 minutes. Remove the thyme sprigs, zest lemon over the top, and season with salt and pepper.