Heat olive oil in a 5-quart Dutch oven over medium-high heat. Sauté the diced onion until translucent and tender, approximately 2 minutes. Incorporate the minced garlic and cook for an additional 30 seconds.

Lower the heat and carefully pour in the hot chicken stock. Add bay leaves and, if desired, the lemon rind. Introduce the halved chicken breasts to the pot, followed by the rice and kosher salt. Allow the mixture to simmer gently for about 15 minutes.

Remove the chicken from the pot and set it aside to cool slightly. Once manageable, shred the chicken into bite-sized pieces using two forks.

Discard the bay leaves and lemon rind from the broth. In a medium-sized bowl, whisk together the eggs, egg yolks, and lemon juice. Gradually ladle in two scoops of the hot broth, one at a time, whisking continuously to temper the eggs without curdling.

Slowly pour the tempered egg mixture back into the soup, stirring consistently to ensure smooth integration. Return the shredded chicken to the pot and let it simmer on low heat for another 5 minutes.

Adjust the seasoning by tasting for salt and lemon. Stir in freshly chopped dill or use it as a garnish. Remove the pot from heat and serve immediately.