Preheat the oven to 400 degrees. If you have a whole chicken, remove the giblets, trim off excess fat, and cut the chicken into 8 pieces (4 breast pieces, 2 thighs, and 2 drumsticks, reserve the wings for another use). In a large bowl, dissolve the salt in 2 quarts of water and immerse the chicken pieces in the brine. Cover and refrigerate for 30 minutes.

Meanwhile, in a small baking dish toss the garlic and shallots with 2 teaspoon of the olive oil and salt & pepper. Cover tightly with foil and roast until softened and beginning to brown, about 30 minutes, shaking the baking dish once about halfway through to toss the contents. Uncover, stir and continue to roast uncovered for another 10 minutes. Remove from the oven and increase the temperature to 450 degrees.

Remove the chicken from the brine, rinse and thoroughly pat dry. Season both sides of the chicken pieces with salt & pepper. Heat the remaining 1 teaspoon olive oil in a 12-inch ovenproof skillet over medium-high heat. Brown the chicken pieces, skin-side down until golden, about 5 minutes. Use tongs to flip the chicken pieces and brown the other side, about 4 more minutes. Transfer the chicken to a plate and discard the fat. Off the heat, add the vermouth, chicken broth, and herbs to the skillet, scraping up any browned bits on the bottom of the pan, and return pan to the heat. Add the garlic-shallot mixture and return the chicken to the pan, nestling the chicken pieces on top of and in between the garlic cloves.

Put the skillet in the oven and roast until a meat thermometer inserted into the thickest part of the breast registers 160 degrees, about 12 minutes. Remove the skillet from the oven and transfer the chicken pieces to a serving dish. Using a slotted spoon, remove & reserve 10 garlic cloves, and scatter the rest of the garlic cloves and shallots among the chicken. Discard the bay leaf and herb stems. Peel the reserved garlic cloves and use a rubber spatula to push them through a fine mesh sieve into the skillet. Bring the liquid to a simmer over medium-high heat, add the butter and whisk to incorporate. Taste and add more salt & pepper if desired. Pour the sauce into a sauceboat and serve with the chicken.