In an oven-safe 10-inch skillet or Dutch oven, over medium-high heat, melt together the 2 tbsp butter and 2 tbsp vegetable oil. Add sausage and sauté for about 6 minutes. Remove the sausage from the skillet/Dutch oven, leaving the butter and oil in the skillet, and set the sausage aside.
In the same skillet/Dutch oven that you cooked the sausage in, add onions, celery, bell pepper and garlic. Sauté until tender, about 5 or 6 minutes. Add tomatoes, chicken stock, hot sauce, oregano, thyme, pepper and brown sugar and reduce heat to medium-low; simmer for about 15 minutes. Season with salt.
Add rice and sausage to the tomato mixture and bake, covered, at (350) for about 55-70 minutes, or until rice is tender. Remove the dish from the oven and allow rice to rest, covered, for 5 more minutes.
(Optional Step) Meanwhile, in a separate skillet, sauté shrimp in 2 Tbsp butter and 2 Tbsp vegetable oil, about 5 minutes. When the shrimp is done and the rice has had a chance to rest, fluff rice with a fork and combine with shrimp in the baking pan.