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.