Chicken breasts are broader at one end than the other, so cut more than halfway up each breast to create two pieces of equal mass. There's no need to take the temperature of the dark meat; it will be properly cooked by the time the white meat reaches its target temperature. If you prefer not to serve the skin, wait until step 6 to remove it; browning the skin produces flavorful compounds that add complexity to the sauce, and braising it releases gelatin, which gives the sauce a rich texture.
1. Dissolve salt in 2 quarts cold water in large container. Submerge chicken in brine, cover, and refrigerate for 30 minutes to 1 hour. Remove chicken from brine and thoroughly pat dry with paper towels. Set aside tapered breast pieces.
2. Adjust oven rack to middle position and heat oven to 300 degrees. Heat oil in Dutch oven over medium-high heat until just smoking. Place all chicken except reserved tapered breast pieces skin side down in pot and cook until skin is well browned, 5 to 8 minutes. (Reduce heat if pot begins to scorch.) Transfer chicken to plate. Pour off all but 2 tablespoons fat from pot, then reduce heat to medium.
3. Add onion and cook, stirring occasionally, until softened, 5 to 7 minutes. Stir in garlic, herbes de Provence, and pepper and cook until fragrant, about 30 seconds. Stir in flour and cook, stirring constantly, for 1 minute. Stir in wine and 1¼ cups water, scraping up any browned bits.
4. Place thighs and drumsticks skin side up in pot and bring to simmer over medium heat. Cover and cook for 8 minutes. (Sauce will have consistency of thick gravy but will thin as chicken cooks.) Add broad breast pieces, skin side down, along with any accumulated juices. Cover and cook until broad breast pieces register 105 to 115 degrees, 3 to 5 minutes. Remove pot from heat.
5. Using tongs, flip broad breast pieces skin side up. Add tapered breast pieces, skin side up, to pot and cover. Transfer pot to oven and cook until breast pieces register 160 to 165 degrees, 15 to 30 minutes.
6. Transfer chicken to serving dish. Discard skin from tapered breast pieces (or all skin, if desired). Sauce should thinly coat back of spoon; if necessary, simmer until slightly thickened, 1 to 2 minutes. Stir parsley and lemon zest and juice into sauce. Season with salt and pepper to taste. Pour sauce over chicken and serve.