In a 6 quart pot or dutch oven, heat the Extra Virgin Olive Oil over medium high heat.  Add the onion and cook 5 minutes, until softened.  Add the garlic and cook for 1 minute, then pour in the chicken broth.  Add the bay leaf, salt, and pepper.  Place the chicken breasts in the pot.  Let the liquid come to a slow boil, then reduce the heat to medium low, cover, and simmer for 20 minutes.

Remove the chicken from the pot and shred.  Add the rice to the pot, then add the shredded chicken back in.  Cover and let simmer for 20 more minutes.  Remove the bay leaf.  

In a bowl, whisk the eggs and lemon juice.  Slowly add 1 cup of broth while whisking, to temper the eggs.  Then add the egg mixture into the pot.  Add the chopped dill and stir to combine.  Let the soup cook for about 5 more minutes, to thicken slightly.