Heat broth and water in a medium saucepan over medium-high heat. Gradually add hot broth mixture and lemon juice to eggs, stirring constantly with a whisk. Return egg mixture to pan. Cook over medium heat until slightly thick (about 15 minutes), stirring constantly.
Remove from heat; stir in rice, salt, and pepper. Ladle soup into each of 6 bowls; serve with lemon slices.