Set the EGG for indirect cooking with a convEGGtor at 300°F/149°C with the Big Green Egg Cast Iron Dutch Oven in the EGG to preheat.
Melt butter in a large Dutch oven over medium-high heat. Once melted, add the garlic and onions and sauté until soft, about 5 minutes. Stir in the tomatoes, chicken stock, barbecue sauce, brown sugar, cayenne, smoked pork pulled pork and/or chicken, lima beans, corn and salt and pepper.
Bring the mixture to a boil, then reduce to a simmer and cook over medium-low heat for 1 1/2 hours, stirring occasionally. Serve with cornbread or biscuits.