Add oil to a large pot heating over medium heat. Saute onion, garlic, carrots, celery, 1/2 teaspoon salt, black pepper, and Italian seasoning until onion is translucent.
Sprinkle flour over vegetables and cook, stirring constantly, for 1 minute.
Remove pot from the heat and slowly whisk in chicken stock until smooth.
Stir in Better Than Bouillon, lemon juice, bay leaf, rosemary sprig, and Parmesan rind and bring to a boil over medium-high heat. Reduce heat immediately and simmer 30 minutes. Remove the bay leaf and Parmesan rind and discard.
Bring the soup back to a high simmer and add the rice and Parmesan cook until it's tender about 15-20 minutes. Add the chicken is heated through, about 5-8 minutes.
Adjust seasoning, ladle into individual bowls and sprinkle with parsley. Serve, passing extra Parmesan.