Preheat the oven to 350 degrees.
In a Dutch oven, melt butter and olive oil over medium low heat. Cube the chicken and seasoning with Cajun seasoning. Cook the chicken until done. Remove the chicken and set aside.
Add celery, onion, white portion of green onions and salt to the Dutch oven. Cook partially covered for 8 minutes, stirring occasionally.
Uncover the vegetables and add garlic. Continue cooking uncovered stirring occasionally for about 8 minutes.
In a separate bowl, combine milk, thyme and cornstarch. Stir until well combined and the mixture is smooth. Let stand for a few minutes.
Once the vegetables are tender and soft, stir the milk mixture vigorously with a fork to make sure there are no clumps. Pour into the vegetable mixture and stir.
Bring to a boil and then reduce heat to a simmer. Add the chicken stock stir. Mix in the fully cooked chicken and uncooked rice.
Pour the mixture from the Dutch oven into the casserole dish. Cover and bake for 1 hour.
Garnish with green onions and serve.