Heat the oil or butter in a large skillet or Dutch oven over medium heat. Add the diced onion, green bell pepper, and celery. Sauté for 5 to 7 minutes, or until the vegetables are tender.

Push the vegetables to one side of the pan. Crumble the ground chicken into the empty space and cook until halfway done. Sprinkle Cajun seasoning over the meat and vegetables. Continue cooking for about 5 minutes until meat is browned.

Add minced garlic and cook for 60 seconds. Pour in chicken broth or water and scrape up browned bits from the pan.

Stir in crushed tomatoes and brown sugar. Bring to a gentle simmer.

Reduce heat to low. Stir in vinegar and any optional additions like hot sauce or heavy cream. Cover and simmer for 10 to 15 minutes to thicken.

Taste and adjust seasoning if needed.

Spoon onto toasted buns and serve immediately with sides like coleslaw or chips.