Preheat oven to 350℉. Brush chicken breasts with olive oil and season with salt and pepper. Bake for 22-25 minutes or until internal temperature reaches 165℉. Set aside to cool and shred.
Meanwhile in a dutch oven or stock pot, heat olive oil over medium heat. Add onion and cook 3-5 minutes until softened. Add garlic, carrots, and celery and cook another 5 minutes.
Add chicken broth and water and bring pot to a simmer over medium high heat and then reduce heat to maintain simmer as needed. Add orzo and cook according to package instructions to al dente. Remove pot from heat.
In a medium bowl, whisk eggs with lemon juice. Slowly pour 1 ladle full of broth to the egg mixture and stir continuously to heat the eggs slowly. We do not want them to scramble! Very slowly, pour the eggs and broth into the soup stirring the soup continuously as you are doing so. Add shredded chicken to pot.
Return pot to low heat. Add a pinch or two of Kosher salt, black pepper, and dill weed along with the handful of chopped greens.