Place strainer over a bowl and empty can of tomatoes into the strainer. Crush (carefully) and reserve tomatoes (set over a dish to contain any drips.) Add the juices in the bowl to a measuring cup (a 4 cup one if you have one) and add enough stock so the total liquid comes up to four cups. Set aside.
Preheat oven to 325°F (160°C). In a Dutch oven, heat oil over medium-high heat until shimmering. Add sausage to Dutch oven and cook, stirring often, until just starting to darken, about 3 minutes. Remove and set aside. Season chicken all over with salt and pepper. Add chicken and cook, turning, until browned on both sides (should be just browned, not cooked through) and remove to the same plate as the sausage.
Lower heat and/or add oil at any point to prevent burning. Add onion, bell pepper, celery, and garlic and cook, stirring and scraping bottom of Dutch oven, until browned bits have come loose and vegetables turn translucent about 10 minutes.
Make a well in the center of the vegetables, add tomato paste and cook, stirring, until lightly browned, about 1 minute. Add hot sauce and Cajun seasoning and allow to heat for seconds, or until just fragrant. Add the reserved crushed tomatoes and tomato/stock mixture and bring to a simmer. Season with salt, tasting liquid to ensure it is well seasoned.
Stir in reserved chicken and sausage, sprinkle rice over top and gently push the rice into the liquid so it is submerged. Return to a simmer, cover with lid and transfer to oven. Bake until liquid is fully absorbed and rice is tender, about 35 to 45 minutes. Remove from oven, covered and let rest 15 minutes. Serve, passing hot sauce at the table for diners to add to taste.
Note: If adding shrimp and or green onions, once the rice is tender, remove the Jambalaya from the oven and gently stir in shrimp and/or green onions. If using green onions only, let the Jambalaya rest for the 15 minutes. If using shrimp, return to oven until shrimp are just cooked through, about 5 minutes, then let it rest for the 15 minutes.
Keywords: Bell Peppers, Cajun & Creole, canned tomatoes, Chicken, Chicken Breast, Chicken Thighs, hot sauce, Jambalaya, Pork, Rice, Sausage, serious eats, smoked sausage