Season the chicken on both sides with a generous amount of salt and pepper. Place a cast-iron or regular ovenproof skillet on the stove over medium heat. Drizzle the pan with the oil and lay the chicken in the hot fat, skin-side down. Cook for about 5 minutes until the skin begins to set and crisp. Throw in the mushrooms, shallots, and rosemary. Stick the whole thing in the hot oven and roast for 15 minutes until the chicken is cooked, and the mushrooms and shallots are soft and roasted.
The last thing to make is a quick sauce using the flavors left in the bottom of the skillet. Take all the stuff out of the pan and arrange on a dinner plate to keep warm while preparing the pan sauce. Pour out all but 1 tablespoon of the rendered chicken fat and return the skillet to the stovetop. Stir in the water and lemon juice and cook over medium heat, scraping up the flavors with a wooden spoon. Cook the liquid down to a syrup, about 5 minutes. Drizzle the pan sauce over the dish, season with salt and pepper.