Preheat oven to 350 degrees F (175 degrees C).

Sprinkle lime juice over the apple slices. Stuff apples into the cavity of the chicken.

Mix 2 cloves garlic, chicken seasoning, thyme, sage, paprika, and mint together in a bowl. Add butter to the garlic mixture; gently toss to coat the butter in the herb mixture. Stuff about 2/3 of the butter pieces under the skin. Coat the outside of the chicken with the remaining butter.

Combine chicken broth, onion, and 2 cloves garlic in a roasting pan. Put chicken onto a rack in the roasting pan with the breast down.

Roast chicken, basting with pan juices every 10 to 15 minutes, in preheated oven for 45 minutes. Flip the chicken and continue cooking until no longer pink at the bone and the juices run clear, about 1 hour and 15 minutes more. An instant-read thermometer inserted into the thickest part of the thigh, near the bone should read 165 degrees F (74 degrees C). Remove the chicken from the oven, cover with a doubled sheet of aluminum foil, and allow to rest in a warm area for 10 minutes before carving.

Pour the juices from the pan into a saucepan and place over medium heat. Stir flour into the juices; cook and stir until the juices thicken into a gravy, about 5 minutes. Serve with the chicken.