• Adjust oven rack to middle position and heat oven to 400°.

Brine the Chicken:

• Dissolve 1/4-cup table salt in 2 quarts cold tap water in large container or bowl.
• Submerge chicken pieces in brine and refrigerate until fully seasoned, about 30 minutes. While the chicken brines, proceed to roast the garlic and shallots.

Roast the Garlic and Shallots:

• Toss garlic and shallots with 2 teaspoons olive oil and salt and pepper to taste in 9-inch pie plate; cover tightly with foil and roast until softened and beginning to brown, about 30 minutes, shaking pan once to toss contents after 15 minutes (foil can be left on during tossing).
• Uncover, stir, and continue to roast, uncovered, until browned and fully tender, 10 minutes longer, stirring once or twice. Remove from oven and increase oven temperature to 450°.

Brown and Roast the Chicken:

• After the chicken has brined for 30 minutes, rinse the chicken pieces under running water and thoroughly pat dry with paper towels. Season both sides of chicken pieces with pepper.
• Using kitchen twine, tie together thyme, rosemary, and bay leaf; set aside.
• Heat remaining 1-teaspoon oil in 12-inch heavy-bottomed ovenproof skillet or Dutch/French oven over medium-high heat until beginning to smoke; swirl to coat pan with oil. Brown chicken pieces skin-side down until deep golden, about 5 minutes; using tongs, turn chicken pieces and brown until golden on second side, about 4 minutes longer. Transfer chicken to large plate and pour off fat to discard.
• Off heat, add wine or vermouth, chicken broth, and herbs, scraping bottom of skillet with wooden spoon to loosen browned bits. Set skillet over medium heat, add garlic/shallot mixture to pan, then return chicken, skin-side up, to pan, nestling pieces on top of and between garlic cloves.
• Place skillet in oven and roast until instant-read thermometer inserted into thickest part of breast registers about 165°, 10 to 12 minutes. If desired, increase heat to broil and broil to crisp skin, 3 to 5 minutes. Using potholders or oven mitts, remove skillet from oven and transfer chicken to serving dish.

Finish Garlic Pan Sauce:

• Remove 10 to 12 garlic cloves to mesh sieve and reserve; using slotted spoon, scatter remaining garlic cloves and shallots around chicken and discard herbs.
• With silicone or rubber spatula, push reserved garlic cloves through sieve and into bowl; discard skins. Add garlic paste to skillet. Bring liquid to simmer over medium-high heat, whisking occasionally to incorporate garlic; adjust seasoning with salt and pepper to taste.
• Whisk in butter; pour sauce into sauceboat and serve.