Season the chicken with 1 1/2 teaspoons salt and a few grinds of pepper. Heat the butter and oil in a large, deep cast-iron skillet over medium-high heat until hot. Add the chicken skin-side down and cook, undisturbed, until deep golden brown, about 5 minutes. Turn the chicken over and turn off the heat. Add the wine, then nestle the olives, thyme, garlic and shallot around the chicken. Return the liquid to a simmer over medium-high heat, then transfer to the oven and roast until the chicken is golden and cooked through, about 1 hour and 15 minutes.
Transfer the chicken, garlic and olives to a serving dish with a slotted spoon. Use a ladle to remove as much fat as possible from the surface of the liquid in the skillet, then pour the liquid over the chicken.
Heirloom garlic, though not as long-lasting as the commercial varieties at the grocery store, has a more intense flavor and cooks up sweeter.