Bring broth to a boil in a five-quart pot, covered, over high heat. Add orzo and simmer briskly, still covered, until al dente (a couple minutes shy of package directions). Meanwhile, in a medium bowl, beat the eggs well with a fork. Beat in lemon juice and set aside.
When orzo is just cooked, ladle out about a cup of hot broth into a measuring cup with a spout. Slowly pour hot broth into bowl with eggs and lemon juice, whisking constantly with a fork. This process tempers the eggs so they don’t scramble when you add them to the pot.
Stir chicken and spinach into soup pot and simmer for a minute or two until spinach is wilted. Reduce heat until broth is no longer bubbling. Stir egg mixture into soup and cook over low heat for a minute or two, until broth looks creamy and is slightly thickened. Add salt and pepper to taste and ladle soup into serving bowls.
If you need to reheat leftovers, do so gently and without boiling.
Use rotisserie or leftover chicken, or quickly sauté a pound of boneless, skinless chicken breasts in a little olive oil while you bring the broth to a boil.