Warm the olive oil in a 5-quart Dutch oven set over medium-high heat. Sauté the diced onion until it becomes translucent, about 2 minutes, then add the minced garlic and cook briefly for 30 seconds.
Reduce the heat and carefully pour in the hot chicken stock. Add the bay leaves and, if preferred, the lemon rind. Place the halved chicken breasts into the pot, followed by the rice and kosher salt. Simmer gently for approximately 15 minutes.
Remove the chicken and allow it to cool slightly before shredding into bite-sized pieces using two forks.
Discard the bay leaves and lemon rind. In a medium bowl, whisk together the eggs, egg yolks, and lemon juice. Gradually ladle in two scoops of hot broth, one at a time, whisking constantly to temper the eggs and prevent curdling.
Slowly return the tempered egg mixture to the pot, stirring steadily to create a smooth, cohesive broth. Reintroduce the shredded chicken and allow the soup to simmer over low heat for an additional 5 minutes.
Taste and adjust seasoning with extra salt or lemon juice as needed. Stir in the fresh dill or use it as a garnish. Serve the soup immediately for best flavor and texture.
