Using a Dutch oven or large soup pot, heat olive oil over medium heat
Add the carrots, onions, and celery and cook/stir for about 8-10 mins.
Add the garlic and cook for another minute.
Next, add the butter to the pan and let it heat to bubbling.When the butter is bubbling, spoon in the flour and stir to create a thick paste. Cook for about 1 minute.
Add the chicken broth in slowly and continue to stir while you bring it to a boil over high heat.
When soup has reached boiling, turn heat down and simmer for about 5 minutes. Vegetables should be tender at this point.
Add milk, salt, pepper, chicken, and Parmesan cheese and continue to let the soup simmer until fully heated and combined. This should only take a few more minutes. Just be sure that the cheese has melted into the soup smoothly.
Serve and enjoy!