Heat the oil in a pot over medium heat. Sauté the onions and garlic for about 5 minutes.
Stir the broth and chicken. Cover with a lid and cook until the chicken reaches an internal temperature of 160°F using a thermometer. Remove the chicken and set aside. Using 2 forks pull the chicken apart to shred.
Add the broccoli florets into the pot and simmer for about 15 minutes. Add the chicken back into the pot. Taste and adjust the seasoning with salt and pepper.
Whisk the flour and milk together until there are no lumps. Slowly add it to the simmering soup; continue to simmer to allow the soup to thicken. Stir in the cheese until it melts completely.
Serve with crusty bread. Garnish with sour cream and a little shredded cheese.