Bring the soup back to a lively simmer. Add the orzo and cook, stirring every minute or so, until the pasta is tender, about 9 minutes. Orzo loves to stick to the bottom of the pot, so this is the moment to stay nearby.
While the orzo cooks, zest the lemons directly into a small bowl, then juice them. Keeping zest and juice ready means you can add lemon off the heat, which protects that bright flavor.
Reduce heat to low. Stir the shredded chicken back into the pot. Add the spinach (if using) and stir just until it wilts, about 1 minute.
Turn off the heat. Stir in the lemon zest, then start with about 1/3 cup lemon juice and add more to taste. (Different lemons vary a lot; you want it clearly lemony, not sour.) Stir in the dill and parsley. Taste and adjust salt and pepper. If the soup tastes flat, it usually needs either more salt or a little more lemon.
Optional egg yolk finish (for a gentle, silky Greek lemon soup feel): In a bowl, whisk 1 egg yolk with 2 tablespoons lemon juice. Slowly whisk in 1/2 cup hot broth from the pot to temper, then pour the mixture back into the soup while stirring constantly. Do not boil after adding, or the egg can curdle. This keeps the recipe in the "brothy" zone, but gives it a soft body.
Serve hot with an extra drizzle of olive oil and more herbs on top. This lemon chicken soup is best right after cooking, when the orzo is perfectly bouncy. If you're planning leftovers, see the FAQ for the best way to keep the pasta from soaking up all the broth.