In a large pot, heat the olive oil over medium heat. Add the chopped onion and garlic, and sauté until softened, about 3–4 minutes.
Pour in the chicken broth and bring to a gentle boil.
Stir in the orzo (or rice) and cook until tender, about 10 minutes.
Lower the heat to a simmer. Stir in the cooked chicken.
In a separate bowl, whisk together the eggs and lemon juice.
Slowly ladle 1 cup of the hot broth into the egg mixture, whisking constantly to temper the eggs.
Slowly pour the egg mixture back into the pot, stirring gently to combine. Do not let it boil.
Season with salt and pepper to taste. Garnish with fresh parsley and serve warm.