1. Dissolve salt in 2 quarts cold water and pat dry with paper towels. Sprinkle with ground pepper. Place oven rack near the bottom and preheat to 450°.
2. Heat oil in heavy 12 inch ovenproof skillet over medium-high heat until it smokes a bit, or about 3 minutes. Place chicken skin side down and fry about 5 minutes. Use a splatter screen or lay a sheet of paper towel over the pan to control spatters without trapping the steam inside and turning the chicken mushy. Turn and brown until golden on the second side, about 4 minutes.
3. Turn chicken skin side down again and place skillet in oven. Roast for about 10 minutes, until thickest part of thighs registers 175° using an instant read thermometer. Remove chicken to platter and let rest while making sauce.
1. Add shallot to pan juices and cook over medium-high heat for about 1 1/2 minutes.
2. Add broth vermouth and thyme. Turn burner to high and simmer until reduced and slightly thickened, about 6 minutes.
3. Discard thyme and whisk in butter. Season with salt and pepper.
4. Return chicken to skillet and simmer to heat through, about 1 minute. Serve immediately.
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