Rinse chicken with cold water and trim off excess fat. Cut chicken into 8 pieces (large breast should be halved so all chicken pieces cook evenly). Dissolve 1/4 cup salt in 2 quarts cold water in large container or bowl. Submerge chicken pieces in brine and refrigerate until fully seasoned, about 30 minutes. Rinse chicken pieces under running water and thoroughly pat dry with paper towels. Liberally season both sides of chicken pieces with pepper.

Meanwhile, separate the cloves of garlic and drop them into a pot of boiling water for 60 seconds. Drain the garlic and peel. Set aside. Tie the parsley, thyme, rosemary, and bay leaf together with string to make a bouquet garni; set aside.

Heat 1 to 2 tablespoons of oil in a deep 12-inch heavy-bottomed skillet or Dutch oven over medium-high heat until beginning to smoke; swirl pan to coat with oil. Add chicken pieces to pan, saute skin-side down until deep golden, about 5 minutes. Use tongs to turn chicken pieces (don't pierce the skin with a fork) and brown second side until golden, about 4 minutes longer. Remove chicken to a platter and tent with foil.

Toss all of the peeled blanched garlic cloves and shallots into the pan and stir. Lower the heat and saute for about 5 to 10 minutes, turning often, until evenly browned. Add 2 tablespoons of the Cognac, vermouth and chicken stock, return to a boil, scraping bottom of pan with wooden spoon to loosen browned bits. Add the bouquet garni, then return chicken to pan, skin-side up, on top of garlic cloves, along with any juices that have collected on the plate. Cover and simmer over low heat for 30 to 40 minutes, or until chicken is cooked through (an instant-read thermometer inserted 2 inches into fleshy part of a thigh registers about 165 degrees), and the gerlic is very soft.

Transfer chicken an to serving platter. Using slotted spoon, remove 10 to 12 garlic cloves to mesh sieve, reserve. Remove remaining garlic cloves and shallots and scatter around chicken, discard bouquet garni, and tent platter with foil. Remove 1/2 cup of the pan sauce to a small bowl, add flour, wisk until smooth. With a rubber spatula push reserved garlic cloves through a sieve and into pan. Whisk flour mixture back into sauce in pan. Raise heat to high, add the remaining tablespoon of Cognac and cream. Boil, whisking to incorporate everything, until sauce thickens, about 3 to 4 minutes. Taste for seasonings, adjusting with salt and pepper to taste. Whisk in butter, if desired. Ladle sauce over chicken, or into a sauceboat to serve on the side.