Heat oil and melt butter in a Dutch oven set over medium-high heat.
To the hot oil, add diced onion and cook for 2 minutes.
Stir in sliced celery and carrots; season with salt and pepper, and continue to cook until tender, about 4 to 5 minutes.
Stir in garlic and cook for 15 seconds, or until fragrant.
Stir in the shredded chicken.
Add oregano, thyme, and rosemary; stir to combine.
Stir in the chicken broth and lemon juice. Add bay leaves.
Increase heat to high and bring to a boil.
Stir in orzo and sliced squash.
Lower heat to a simmer and cook for 10 minutes, or until orzo is tender.
Taste soup for salt and pepper; adjust accordingly.
Ladle soup into bowls.
Garnish with parsley and serve with lemon wedges.