In a large pot over medium heat, warm the olive oil. Add onion, ½ teaspoon salt, and black pepper and cook until softened, about 5 minutes.
Stir in the rice and/or quinoa, bay leaf (if using), chicken, water, and 2 teaspoons salt. Bring to a boil, then reduce the heat to medium-low to maintain a rapid simmer/low boil for 25 minutes uncovered. We want the liquid to reduce and flavor from the chicken bones to infuse the broth. If using brown rice, be sure to taste and make sure it is cooked through.
Turn the heat off, remove the bay leaf (if using) and transfer the chicken to a cutting board. Use two forks to shred the chicken. Return the chicken to the pot along with the spinach (if using). Turn the heat to medium low and simmer until the spinach wilts, about 2 minutes.
In a medium bowl, whisk together the 2 whole eggs, 2 egg yolks, and lemon juice until smooth. VERY slowly pour in a scant 2 cups of the hot broth while whisking constantly (it’s ok if you have some bits of food in the broth!).
Stir the lemony egg liquid back into the soup. Simmer it gently for a couple of minutes. It will become creamy — MAGIC! Taste and add remaining ½ teaspoon more salt if you think it needs it (the acidic broth really needs salt to balance it out!).
Serve topped with lots of fresh dill, a drizzle of olive oil and lots of black pepper.