Preheat oven to 325 degrees F (165 degrees C). Season roast on all sides with salt and pepper.
Heat oil in a Dutch oven over high heat. Brown roast on all sides starting with the meat side down; cook until well browned, 4 to 5 minutes. Flip over and cook until the bone side is browned, 4 to 5 minutes. While second side browns, place chopped onions and garlic around the roast. As roast browns, the onions will begin to turn translucent. Add chicken broth, cider vinegar, honey, and red pepper flakes. Stir together, turning roast to coat all sides.
With meat side up, cover pot and place on center rack in preheated oven. Roast until meat is fork tender, about 3 hours.
Spoon off excess fat. Break up meat into small strips and place in a pot. Discard bones. Strain broth into pot with picked meat. Place pot over medium heat and cook until simmering. Stir in chopped fresh mint.
