In a large Dutch oven, heat olive oil and butter over medium heat. Add garlic, onions, and celery. Sauté for a few minutes, until the onions are soft and translucent.
Add the broccoli florets, chicken breasts, chicken broth, oregano, parsley, salt, and pepper to the pot. Bring to a boil, then lower the heat and simmer for about 20 minutes.
Remove the chicken breasts from the pot and shred them with a fork. Set aside.
Using an immersion blender (or a regular blender), blend half of the soup until smooth.
Return the shredded chicken to the pot, then stir in the cream cheese and heavy cream. Let the soup simmer for an additional 5 minutes. Taste and adjust the seasoning as needed.
Serve hot and enjoy!