In a large soup pot or Dutch oven, heat the olive oil over medium heat. Add the diced carrots, celery, and onion. Sauté for 5-7 minutes until the vegetables soften. Add the minced garlic and cook for 1 more minute until fragrant.
Pour in the chicken broth, the rice blend, the contents of the rice seasoning packet, the cooked chicken, dried thyme, and black pepper. Stir well to combine.
Bring the mixture to a boil, then reduce the heat to low. Cover and simmer for 15-20 minutes, or until the rice is tender.
While the soup simmers, prepare the roux. In a separate medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until bubbly and golden (do not burn).
Slowly whisk the milk into the butter and flour mixture a little at a time to prevent lumps. Continue whisking and cooking until the mixture thickens into a creamy white sauce (about 3-4 minutes).
Pour the creamy white sauce into the large soup pot with the rice and chicken. Stir gently to combine.
Simmer the combined soup for another 5-10 minutes uncovered, allowing it to thicken and the flavors to meld. Taste and adjust salt and pepper if needed before serving.