Fill a Dutch oven or stockpot about water or stock; mix chicken base, tarragon, basil, bay leaves, and black pepper into water and bring to a boil. Cook chicken in the boiling broth mixture until no longer pink in the center, 15 to 20 minutes. Remove chicken from broth, leaving broth in Dutch oven to simmer. Dice chicken and add it back to the simmering broth.
Mix rice, onion, carrot, celery, and garlic into simmering broth; cook until rice is tender and soup is thickened, 25 to 30 minutes. Add more water if needed. Add salt to taste, but bouillon and stock is already pretty salty.
Creamy:
In a medium saucepan over medium heat, melt butter. Reduce heat to low, then stir in flour mixture by tablespoons, to form a roux. Whisk in cream, a little at a time, until fully incorporated and smooth. Whisk in milk similarly. Cook until thickened, 5 minutes. Stir into soup.