Step 1
Place a rack in middle of oven; preheat to 350°. Pat 2 lb. skin-on, bone-in chicken thighs (4–6) dry with paper towels and sprinkle all over with 1 tsp. freshly ground pepper and 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt.
Step 2
Pour 3 Tbsp. extra-virgin olive oil into a cold large high-sided ovenproof skillet or short-sided pot over medium (you want the vessel to be at least 3½-qt. capacity; if you don’t have any of the preferred options, a medium Dutch oven will work). Arrange chicken thighs, skin side down, in pan and set over medium heat. Cook, undisturbed, rotating pan around on burner for even browning, until chicken is deep golden brown and releases easily from pan, 14–17 minutes. Transfer chicken thighs to a plate, arranging skin side up.
Step 3
Pour off all but 4 tsp. oil from pan; discard. Return pan to medium heat and cook 3 medium shallots, thinly sliced, stirring often, until just tender, about 4 minutes. Add 12 garlic cloves, thinly sliced, and cook, stirring often, until fragrant, about 3 minutes. Sprinkle in 1 Tbsp. all-purpose flour and stir to coat, then remove from heat (make sure pan is far away from the burner).
Step 4
Carefully pour in 1¼ cups dry vermouth, 1 cup low-sodium chicken broth, ⅔ cup olive brine, and ⅓ cup vodka. Place pan over medium-high heat and bring liquid to a simmer. Cook, stirring occasionally, until sauce is reduced by two thirds, 9–12 minutes. Remove from heat; stir in 1 tsp. finely grated lemon zest and remaining ⅓ cup vodka. Taste sauce and season with more salt if needed.
Step 5
Nestle chicken thighs, skin side up, into sauce. Scatter one 5.5-oz. jar pitted Castelvetrano or other green olives, rinsed (about 1 cup), over, allowing most of them to fall into the sauce. Bake, uncovered, until chicken is cooked through and an instant-read thermometer inserted into the thickest part of a thigh registers 165°, 25–35 minutes.
Step 6
Remove pan from heat and scatter parsley leaves with tender stems over chicken and sauce. Serve with lemon wedges for squeezing over if desired.