Preheat oven to 425ºF.
Heat a large, ovenproof skillet with oil, 2 turns of the pan, over medium-high heat. Add the celery, carrots, onions, garlic, chile pepper, salt, pepper and thyme, and sauté to soften the vegetables, 10-12 minutes. Deglaze the pan with the beer or stock.
Stir together the brown sugar, tomato sauce; hot sauce, Worcestershire and vinegar in a bowl. Pour into the skillet and stir to combine. Fold in the chicken. (Let cool and store for a make-ahead meal. Then reheat over medium heat just to warm up while you mix together the cornbread topper.)