There's so much going on in our version of Greek cream-of-chicken soup with rice: Eggs enrich the chicken broth, lemon gives it tang, and all of it is backdrop to the peppery watercress, stirred in at the very end so it stays a vibrant green. And if you have it on hand...Stir in some leftover cooked, diced chicken when you add the watercress.
Bring the chicken broth to a boil in a medium saucepan, covered, over high heat. Stir in the rice, some salt to taste and a bit of black pepper. Cover, lower the heat to medium-low and boil until the rice is just cooked through, about 15 minutes.
When the rice is just about done, beat the eggs with a whisk in a medium bowl. Whisk in the lemon juice.
Whisking constantly, slowly ladle about 2 cups of the hot broth into the eggs. Lower the heat under the remaining broth to low and whisk the egg mixture into the pan. Stir with a wooden spoon until the soup thickens slightly, about 2 minutes. Do not let the soup boil.
Remove the soup from the heat and stir in the lemon zest and watercress; let sit until the watercress is tender but still vibrant green, 1 to 2 minutes. Ladle into 6 bowls and sprinkle each serving with a bit of black pepper.