Preheat oven to 450F. Grease heavy large rimmed baking sheet. Pat chicken dry with paper towels. Starting at edge of cavity, slide finger, under skin over each breast half, making pockets. Place 1 sprig tarragon and 1 sprig thyme in each pocket. Sprinkle chicken with 1 1/2 tsp. each minced tarragon and thyme; season with 3/4 tsp salt and 1/4 tsp pepper. Place remaining 2 sprigs tarragon and 2 sprigs thyme in cavity. Place chicken on baking sheet. Roast 30 minutes
Meanwhile, in large bowl, toss potatoes and carrots with oil, 2 tsp. each minced tarragon and thyme and remaining salt and pepper. After chicken has roasted 30 minutes, remove from oven. Tilt pan and spoon off most of fat. Arrange chicken in center of pan; spoon vegetables around bird; turning potatoes cut side down.
Return pan to oven. Roast until chicken juices run clear when meat is pierced with fork, 30-40 minutes longer. Transfer chicken and vegetables to platter to rest. Pour pan juices into pot; skim fat.
Add broth to baking sheet; set over medium heat. Bring to boil; stirring up any browned bits; pour into pot. Boil until reduced slightly, about 5 minutes; add any juices from chicken platter. Mix remaining tarragon and thyme into sauce. Serve with chicken and vegetables.