Heat oil in a 6 quarts Dutch oven pot over medium-high heat. Season the chicken generously with salt and pepper, all sides. Add to pot and cook undisturbed until skin is golden brown, and you can turn the pieces over with no difficulties. Turn only once.
Add garlic to the pot and cook until browned in some spots, about 5 minutes. Add vermouth and cook, scraping the bottom of the pot, until slightly reduced, about 2 minutes. Add the chicken stock and bring it to a boil. Transfer half the garlic cloves to the baking dish. Mash the remaining garlic cloves into stock and pour over the chicken.
Bake until glazed and tender, 25 to 30 minutes. Garnish with tarragon and serve.