1 whole chicken, cut into 8 serving pieces (I bought a pre-cut chicken at the store)
Ground black pepper
3 large heads garlic, with medium-sized cloves if possible
2 medium shallots, peeled and quartered pole to pole
3 tsp. olive oil, divided
2 sprigs fresh thyme
1 sprig fresh rosemary
1 bay leaf
¾ cup dry vermouth or dry white wine
¾ cup low-sodium chicken broth
2 tbsp. unsalted butter
Adjust an oven rack to middle position. Preheat the oven to 400° F. In a large container or bowl combine ¼ cup salt in 2 quarts of cold tap water; stir until the salt is dissolved. Submerge the chicken pieces in the brine and refrigerate until fully seasoned, about 30 minutes. Rinse the chicken pieces off and pat dry with paper towels. Season the chicken pieces on both sides with freshly ground pepper.
Remove the outer paper skins of the heads of garlic. Separate the cloves but keep the peels on. In a 9-inch pie plate combine the garlic cloves and quartered shallots. Toss with 2 teaspoons olive oil and season with salt and pepper to taste. Cover tightly with foil and roast until softened and beginning to brown, about 30 minutes, shaking the pan once to toss the contents after 15 minutes (leave the foil on to toss). After 3o minutes, remove the foil and return the pan to the oven to bake uncovered for 10 more minutes, so the cloves are fully browned and tender. Remove from the oven and increase the oven temperature to 450° F.
Using kitchen twine, tie together the fresh thyme, rosemary and the bay leaf; set aside. Heat the remaining 1 teaspoon olive oil in a 12-inch skillet or Dutch oven over medium-high heat until beginning to smoke; swirl to coat the pan with oil. Brown the chicken pieces skin side down until deep golden, about 5 minutes. Turn the chicken pieces with tongs and cook until golden brown on the second side, about 4 minutes longer. Transfer the chicken to a large plate and discard the fat. Off the heat, add the vermouth or white wine, chicken broth, and herb bundle, scraping the bottom of the pan to loosen the browned bits. Set the skillet over medium heat, add the garlic-shallot mixture to the pan. Return the chicken pieces to the pan, skin side up, and nestle them on top of and between the garlic cloves.
Place the pan in the oven and roast until an instant-read thermometer inserted into the thickest part of the breast registers 160° F, about 10-12 minutes (mine took about 18 minutes). If desired, increase the heat to broil and broil to crisp the skin, about 3-5 minutes more. Remove the pan from the oven and transfer the chicken to a serving dish. Reserve 10-12 cloves of garlic and place them in a mesh sieve. Using a slotted spoon, scatter the remaining garlic cloves and shallots around the chicken and discard the herbs. Push the reserved garlic cloves through the mesh sieve into a bowl; discard the skins. Add the garlic paste to the pan and bring the liquid to a simmer over medium-high heat, whisking occasionally to incorporate the garlic. Adjust seasoning with salt and pepper to taste. Whisk in the butter. Transfer to a sauce dish and serve with chicken pieces.