Heat the oven to 375°F and grease a 9-by-13-inch baking dish.
Brown the sausage in the vegetable oil over medium-high heat for 4 minutes and transfer it to a bowl.
Cook the onion, bell pepper and celery in the same skillet until softened, about 6 minutes, then stir in the garlic.
Combine the sausage, vegetables, cooked chicken, rice, tomatoes, broth, Cajun seasoning and salt in the baking dish.
Cover with foil and bake until steaming throughout, 25 minutes.
Remove the foil, sprinkle on the Monterey Jack and bake until melted, about 8 minutes.