Pour chicken broth into a large stockpot or Dutch oven over medium-high heat. Add quartered onion, chopped carrots, celery with leaves, bay leaves, dried oregano, salt, and pepper. Bring to a gentle boil, about 5-7 minutes.
Add chicken to boiling broth, reduce heat to medium-low, and cover with lid slightly ajar. Simmer for 20-25 minutes for breasts or 25-30 minutes for thighs until chicken reaches 165°F (74°C). Remove chicken and shred with two forks.
Strain broth through a fine-mesh strainer into a large bowl, discarding vegetables and bay leaves. Return clear broth to pot and bring to a boil. Taste and adjust salt if needed.
Add rice or orzo to boiling broth and stir. Reduce heat to medium and cook uncovered, stirring occasionally, for 15-18 minutes for rice or 8-10 minutes for orzo. Remove from heat and let cool slightly for 2 minutes.
Crack eggs into a medium bowl and whisk vigorously for 2-3 minutes until pale yellow, frothy, and doubled in volume. Add fresh lemon juice and lemon zest (if using) and whisk another minute until smooth.
Remove soup pot from heat completely. While whisking egg mixture constantly, slowly drizzle in 1 cup of hot broth in a thin stream. Repeat two more times, adding 1 cup of hot broth each time while whisking continuously.
Slowly pour the tempered egg-lemon mixture into the soup while stirring gently with a wooden spoon for 1-2 minutes as soup thickens. Add shredded chicken back to pot and stir to distribute. Do not return to a boil.
Ladle soup into warmed bowls. Garnish with fresh chopped dill or parsley, drizzle with extra virgin olive oil, and add fresh black pepper. Serve immediately with lemon wedges on the side.