Cook rice according to instructions on package.
While rice is cooking, heat 2 Tablespoons of oil, butter or ghee in a Dutch oven or large pot. Once hot add onion, carrots and celery and saute for 5-6 minutes, stirring occasionally. Add garlic and saute for 1 minute longer.
Add thyme, rosemary, salt and bay leaves and mix until veggies are coated.
Add chicken broth, water and chicken into the pot. Bring the mixture to a boil, reduce heat to a simmer and cover. Simmer for about 20 minutes or until chicken is fully cooked. When chicken is fully cooked, remove from pot and shred using two forks.
Add shredded chicken and cooked rice into pot.
In a separate saucepan melt ¼ cup oil, butter or ghee. Add the flour and let the mixture cook for 1 minute. Whisk the mixture slowly while adding in the milk. Continue to whisk until all lumps have dissolved. Allow the mixture to thicken and become creamy.
Add the creamy mixture into the pot. Stir to combine. Add additional water or milk to your preference if the consistency is too thick. Taste and season with salt and pepper if needed. Garnish with fresh thyme and serve.