In a large pot or Dutch oven, brown the sausage and chicken over medium heat. Remove from the pot and set aside.
In the same pot, melt the butter. Whisk in the gluten-free flour and cook for 2-3 minutes, stirring constantly, until the roux turns a light golden brown.
Add the chopped onion, celery, and bell pepper to the roux. Cook and stir until softened, about 5-7 minutes.
Stir in the minced garlic, Cajun seasoning, salt, and pepper. Cook for 1 minute more.
Add the diced tomatoes (with their juices) and chicken bone broth. Bring to a boil.
Stir in the jasmine rice. Reduce heat to low, cover, and simmer for 20-30 minutes, or until the rice is tender and the liquid is absorbed.
Return the browned sausage and chicken to the pot.
Stir gently to combine. Serve hot.