In a large, heavy-bottom pot (I use a large Corning Ware), heat 1 tablespoon olive oil on medium heat. Once hot, add all vegetables and sauté for about 2 minutes. Add broth and decrease to medium-low heat. Add chicken breasts, piercing the chicken multiple times with a knife, and 1 cup uncooked rice. Stir occasionally throughout cook to avoid sticking.
Add 1 teaspoon sea salt, freshly ground pepper to taste, 1 ½ teaspoon dried dill, and 1 bay leaf.
Cook for about 13-15 minutes, flip chicken, then decrease heat to low. Cook for another 28-30 minutes, then shred the chicken in the pot using 2 forks. Continue cooking for 20 more minutes.
After about 15 minutes of the cook time is up, begin next step: separate 2 large eggs and whisk each separately, then whisk them together, along with ¼ cup lemon juice. Temper the eggs by slowly adding in 1 cup of hot broth from the soup pot into the eggs while whisking continuously (important step). Add 1 additional cup of broth, whisking again.
After the 20 minutes is up, reincorporate the tempered eggs back into the soup pot, stirring continuously to avoid scrambling the eggs.
Continue to cook for another minute or two on simmer then remove from heat.