Add olive oil and 1 tablespoon butter to a large dutch oven or stockpot and heat over medium heat. Season chicken breasts on both sides with salt and pepper. Add chicken to dutch oven and cook 3-4 minutes per side, until golden brown and cooked through. Set chicken aside on a plate.
Add remaining butter to dutch oven, then add onion, carrots, celery and garlic. Cook, stirring often, until soft, about 3 minutes. Stir in dried thyme.
Pour in chicken broth and water, using a wooden spoon to scrape up any browned bits in the bottom of the dutch oven. Bring to a boil.
While soup is coming up to a boil, shred chicken using two forks. Once soup is starting to boil, stir in shredded chicken, orzo, bay leaf and add sprigs of rosemary and oregano. Reduce heat and simmer 10-12 minutes, until orzo is tender.
Remove and discard bay leaf and rosemary/oregano sprigs. Stir in lemon juice and fresh parsley. Taste and adjust seasonings if necessary.