Pat the chicken thighs dry, and season with salt and pepper.
Heat 1 tablespoon extra-virgin olive oil in a Dutch oven over medium-high heat, add the chicken and brown on both sides in 2 batches.
Remove the chicken to a plate and spoon out 1/2 the drippings, and add the andouille sausage. Brown for 2 minutes and then add the onion, celery, pepper, garlic, and thyme, and cook to soften, for about 10 minutes over medium heat.
Add the flour, stir 1 to 2 minutes, and then pour in the beer and let the foam subside. Stir in the tomatoes, stock, and hot sauce. Let the sauce thicken a bit, and then slide the chicken into the pot and simmer to cook through. Serve with warm crusty bread, or cool and store for make-ahead meal.