Warm the olive oil in a large stock pot or Dutch oven over medium heat, sautéing the diced onion, carrot, and celery until softened, about 8-10 minutes. Introduce the finely chopped garlic and cook for an additional minute. Add the bay leaves, chicken breasts, and chicken stock, then bring the mixture to a boil.
Lower the heat to medium-low, cover with a lid, and let it simmer for 15 minutes. Remove the chicken breasts and bay leaves from the pot. Add the orzo to the soup and cook for about 10 minutes.
Using two forks, shred the chicken and return it to the pot. Meanwhile, in a separate bowl, whisk together the egg yolks and freshly squeezed lemon juice. Gradually temper this mixture by whisking in 1-2 ladles of hot soup. Slowly stir the tempered egg-lemon mixture back into the pot, ensuring the soup does not boil, and cook for an additional 5 minutes.
Fold in the fresh dill, adjust the seasoning with salt and pepper, and serve the soup immediately, garnished with extra dill if desired.