Heat EVOO in a large Dutch oven pot and cook the andouille sausage for about 2 minutes. Add the ground chicken and pork in an even layer; do not overcrowd the pot. Break up the meat with the back of a wooden spoon and scrape up the bits at the bottom. Continue to cook until all the meat is super-duper brown. To the pot, add the celery, onion, green bell pepper and garlic, and cook for 3-4 minutes.
Add the hot sauce, thyme, chili powder, cumin, paprika and cayenne, and continue cooking for another minute. Pour in the stock, crushed tomatoes and tomato sauce, bring to a bubble and turn down to a simmer. Let cook for another 20 minutes or until reduced.