In a large stock pot, over medium heat, melt the butter and oil. Add the onions, carrots, celery and garlic to the pot. Saute until soft, about 3-5 minutes.
Add the uncooked rice to the pot and stir to cover in butter/oil.
Sprinkle flour over vegetable/rice mixture. Stir to combine and let cook for for about 2-3 minutes until the flour is no longer raw.
Season the chicken with salt and pepper. Add the chicken and bay leaf to pot.
Slowly add the chicken stock and bring to a boil and then lower the temperature to low. Cover and let simmer for about 40 minutes or until the rice is tender. Stir several times throughout the cooking process to ensure that the rice does not stick to the bottom of the pot.
Once the rice is cooked through, remove the lid and stir in the milk and cream. Season with salt and pepper.
Stir in the fresh herbs and let simmer on low for about 5 minutes for the flavors to incorporate.