Heat olive oil in a large Dutch oven or pot over medium heat. Add the diced onion and cook for 3-4 minutes until softened. Add the garlic and cook for another 30 seconds until fragrant.

Pour in the chicken broth and bring to a boil. Add the chicken breasts, reduce heat to medium-low, and simmer for 15-18 minutes until the chicken is cooked through (internal temperature of 165°F). Remove chicken to a cutting board and shred with two forks.

While chicken rests, add the rice to the simmering broth. Cook for 15-18 minutes until the rice is tender, stirring occasionally.

In a medium bowl, whisk together the eggs, lemon juice, and lemon zest until well combined and slightly frothy.

Reduce the soup heat to low. Very slowly ladle about 1 cup of hot broth into the egg mixture, whisking constantly. Then add another cup, continuing to whisk. This tempers the eggs so they won't scramble.

Slowly pour the tempered egg mixture back into the pot while stirring constantly. Add the shredded chicken and fresh dill. Stir gently and cook on low for 2-3 minutes until slightly thickened. Do not boil.

Taste and season with salt and pepper as needed. Ladle into bowls and garnish with extra fresh dill and a drizzle of olive oil if desired. Serve immediately.