Cook the rice according to package directions.
Meanwhile, add the olive oil, butter, celery, carrots, and onions to a Dutch oven or soup pot over medium-high heat. Sauté for 7-8 minutes, stirring occasionally, or until softened and lightly browned. If it starts to brown too much, turn the heat down a bit.
Stir in the garlic and flour and cook for about a minute.
Slowly pour in the broth and stir/scrape up the browned bits from the bottom of the pot until the flour has dissolved.
Stir in the Italian seasoning, cream, chicken, and cooked rice. Increase the heat to high, and once the soup comes to a gentle boil, give it a good stir, then reduce the heat so it's simmering and cover the pot with the lid slightly open. Simmer the soup for 8-10 minutes.
Season with salt & pepper as needed (I am generous with both) and stir in the chopped parsley if using.