In a large pot, bring the broth, water, and 1 teaspoon of the salt to a simmer. Add the chicken breasts to the pot and simmer until they are just done, about 10 minutes. Remove the chicken breasts from the pot; when they are cool enough to handle, cut them into bite-size pieces.
Meanwhile, stir the rice into the simmering broth. Increase the heat and boil until the rice is almost tender, about 10 minutes. Reduce the heat to the lowest possible temperature.
In a medium glass or stainless-steel bowl, beat the eggs, lemon juice, and the remaining 1/2 teaspoon salt until frothy. Remove about 1 cup of the hot broth from the pot. Pour the hot liquid in a thin stream into the egg mixture, whisking constantly. Pour the mixture back into the pot and stir until the soup begins to thicken, about 3 minutes. Do not let the soup come to a simmer, or it may curdle. Return the chicken to the pot and stir in the peas.
Stir in three tablespoons of chopped fresh dill along with the peas.
Wine Recommendation: A wine really needs to be rich to hold its own with the egg thickener here, yet high enough in acidity to be refreshing. Though expensive, a Meursault, one of the great white Burgundies, will pair with the soup to make an experience you won't forget.