Rinse the chicken inside and out and pat it dry with paper towels. Rub the chicken with 1 tablespoon of the oil and season with 1 1/2 teaspoons of the salt and 1/4 teaspoon of the pepper. Place in a roasting pan or baking dish.
In a small bowl, combine the peaches, shallots, vinegar, thyme, and the remaining oil, salt, and pepper. Scatter the peach mixture around the chicken. Roast until the chicken is cooked through and a thigh registers 180° F on a meat thermometer, about 1 hour, 10 minutes. Let rest for 10 minutes before carving. Serve with the peaches and pan sauce.
Tip: If you can't find fresh peaches, substitute one 16-ounce bag of thawed frozen peaches. Quarter, rather than halve, the shallots, and add the shallot-peach mixture to the pan after the chicken has been in the oven for 30 minutes.