Combine the first 3 ingredients in a small bowl to make the Seasoning Mix.
Melt the butter in a 2-quart saucepan over high heat. Add the tasso and cook, stirring frequently, until the meat starts to brown, about 3 minutes. Add the chicken and continue cooking, stirring frequently and scraping the pan bottom well, until the chicken is browned, about 3-5 minutes. Stir in the seasoning mix, ½ cup each of the onions, celery, and bell peppers, and the garlic.
Continue to cook the mixture, stirring almost constantly and scraping the pan bottom as needed, until the vegetables start to get tender, about 5-8 minutes. Stir in the tomato sauce and cook, stirring often, for 1 minute. Stir in the remaining onions, celery, and bell peppers and the tomatoes. Add the stock and rice. Bring to a boil then reduce heat to low. Cover and simmer until the stock is absorbed and the rice is cooked, about 20 minutes. Let sit for 5 minutes before serving. Sprinkle on chopped green onions or scallions. Sauce Piquant works well as a sauce served underneath the Jambalaya.