Combine the chicken broth, bone broth, lemon zest, juice of 1 lemon, bay leaves, salt, and black pepper in a large saucepan over medium heat. Simmer for 10-15 minutes.

In a large bowl, whisk the eggs until smooth. Whisk in the juice of half the remaining lemon, then the other half.

Add a ladle of the hot broth to the eggs while whisking continuously for 1 minute. Repeat this two more times to temper the eggs.

Gradually pour the egg mixture into the hot broth while whisking continuously. Do not bring to a boil at this point or the eggs could curdle. Heat until warmed through.

Remove from heat and stir in the shredded chicken and cooked rice.

Serve warm with fresh dill, a drizzle of high-quality extra virgin olive oil, and freshly cracked black pepper.