Heat the olive oil in a large pot over medium heat. Add the onion, carrots, and celery. Cook for 5 minutes until the vegetables begin to soften.
Stir in the garlic and cook for 1 minute more.
Add the chicken breasts and chicken broth to the pot. Bring the mixture to a boil, then lower the heat and simmer for 15 minutes.
Remove the chicken breasts from the pot and set them aside. Add the rice to the simmering broth. Cook for 15 minutes until the rice is tender.
Shred the chicken with two forks and return it to the pot. Stir in the lemon juice and fresh dill.
Season with salt and black pepper. Serve the soup hot.