Put oven rack in upper third of oven and preheat oven to 500 degrees F.
Cook garlic, thyme, and 3/4 cup oil in a 1 to 1 1/2-quart heavy saucepan over low heat, uncovered, until garlic is very tender but not golden, about 25 minutes.
While garlic is cooking, pat chicken pieces dry and rub all over with butter and remaining tablespoon oil. Sprinkle all over with salt and pepper. Arrange chicken, skin sides up, in a shallow baking pan and roast 20 minutes.
Transfer garlic to a small bowl along with 1 tablespoon garlic oil and mash with a fork. Spread mashed garlic over skin of roasted chicken, then return chicken to oven and roast until just cooked through and skin is crisp, about 5 minutes.