For the chicken:
½ cup salt
2 quarts water
2 each bone-in, skin-on chicken breasts and thighs, breasts cut into four pieces
Freshly ground black pepper
1 tablespoon vegetable oil
For the sauce:
1 large shallot, minced
¾ cup low-sodium chicken broth
¼ cup white wine
¼ cup cognac or brandy
2 sprigs fresh thyme, plus more for garnish
1 tablespoon Dijon mustard
2 tablespoons unsalted butter, cut into tablespoon-sized pieces and chilled
Salt and pepper, to taste
Dissolve salt in cold water in a large container. Submerge the chicken in brine and refrigerate for 30-60 minutes (skip if using a kosher chicken). Remove from brine and pat dry with paper towels. Season generously with freshly ground black pepper.
Place oven rack on lowest position and heat oven to 450º. Heat oil in a large ovensafe skillet over medium high heat until just smoking. Place the chicken, skin side down, into the skillet and cook for 6-8 minutes, or until the chicken is well browned, reducing the heat as needed. Flip and continue cooking for an additional 3 minutes, until the chicken is lightly browned on the second side.
Flip the chicken skin side down again and place the skillet in the oven. Cook for about 10 more minutes, until the chicken is cooked through. Remove the skillet from the oven and transfer chicken to a serving platter. Pour off all but about a teaspoon of fat left in the skillet.
Place the skillet over medium-high heat and add the shallot. Cook for about 2 minutes, until softened. Stir in the chicken broth, white wine, cognac or brandy and thyme sprigs. Use a whisk to scrape up any browned bits from the bottom of the pan. Bring to a simmer and cook until slightly thickened, about 6 minutes. Stir in any accumulated chicken juices and continue cooking another 30 seconds. Remove from heat and remove the thyme sprigs. Whisk in the Dijon and then the butter, 1 piece at a time. Taste and add salt and pepper as needed. Pour sauce over chicken and serve immediately, sprinkling with more fresh thyme if desired.