Step 1
Place a rack in center of oven; preheat to 350°. Pat 2 lb. skin-on, bone-in hen thighs (4–6) dry with paper towels and sprinkle throughout with 1 tsp. freshly floor pepper and 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt.
Step 2
Pour 3 Tbsp. extra-virgin olive oil into a chilly giant high-sided ovenproof skillet or short-sided pot over medium (you need the vessel to be at the very least 3½-qt. capability; in case you don’t have any of the popular choices, a medium Dutch oven will work). Prepare hen thighs, pores and skin facet down, in pan and set over medium warmth. Cook dinner, undisturbed, rotating pan round on burner for even browning, till hen is deep golden brown and releases simply from pan, 14–17 minutes. Switch hen thighs to a plate, arranging pores and skin facet up.
Step 3
Pour off all however 4 tsp. oil from pan; discard. Return pan to medium warmth and prepare dinner 3 medium shallots, thinly sliced, stirring typically, till simply tender, about 4 minutes. Add 12 garlic cloves, thinly sliced, and prepare dinner, stirring typically, till aromatic, about 3 minutes. Sprinkle in 1 Tbsp. all-purpose flour and stir to coat, then take away from warmth (be certain that pan is much away from the burner).
Step 4
Fastidiously pour in 1¼ cups dry vermouth, 1 cup low-sodium hen broth, ⅔ cup olive brine, and ⅓ cup vodka. Place pan over medium-high warmth and convey liquid to a simmer. Cook dinner, stirring sometimes, till sauce is diminished by two thirds, 9–12 minutes. Take away from warmth; stir in 1 tsp. finely grated lemon zest and remaining ⅓ cup vodka. Style sauce and season with extra salt if wanted.
Step 5
Nestle hen thighs, pores and skin facet up, into sauce. Scatter one 5.5-oz. jar pitted Castelvetrano or different inexperienced olives, rinsed (about 1 cup), over, permitting most of them to fall into the sauce. Bake, uncovered, till hen is cooked by means of and an instant-read thermometer inserted into the thickest a part of a thigh registers 165°, 25–35 minutes.
Step 6
Take away pan from warmth and scatter parsley leaves with tender stems over hen and sauce. Serve with lemon wedges for squeezing over if desired.
