Sauté the aromatics: Heat olive oil and butter in a large pot over medium heat.
Add onion, carrots, and celery. Cook, stirring occasionally, until softened, about 6–8 minutes. Add garlic and cook 30 seconds until fragrant.
Season the base: Stir in salt, pepper, oregano, and bay leaf.
Let the spices bloom for 30 seconds to deepen the flavor.
Add broth and chicken: Pour in the chicken broth and bring to a gentle boil. Nestle in the chicken thighs. Reduce to a simmer, cover, and cook until the chicken is cooked through and tender, 15–20 minutes.
Shred the chicken: Transfer the cooked chicken to a cutting board and shred with two forks.
Discard any bay leaf in the pot.
Cook the orzo: Bring the soup back to a gentle boil and add the orzo. Cook uncovered, stirring occasionally so it doesn’t stick, 7–9 minutes or until al dente.
Finish the soup: Return shredded chicken to the pot. Add lemon zest and juice.
Taste and adjust salt and pepper. If using, stir in a splash of heavy cream or temper in the beaten egg for extra body. Remove from heat.
Herbs and serve: Stir in dill or parsley just before serving.
Ladle into bowls and serve with extra lemon wedges.