In a large saucepan, bring the chicken stock, water, and rice to a simmer. Cook until the rice is tender, about 20 minutes.
In a small bowl, beat the eggs until light yellow. Whisk a few tablespoons of the soup into the eggs. Then slowly add the egg mixture to the soup, whisking constantly to prevent the eggs from scrambling. Cook over medium heat, whisking constantly, until the soup reaches 160 degrees F, about 5 minutes. Add in the lemon juice and soy sauce, stir to combine, and serve garnished with the green onions and freshly ground black pepper, if desired.