First, make the rice: In a three-quart saucepan, bring the water to a boil. Add the rice and salt and cover and simmer it for 16 minutes. Remove it from the stove and let it sit covered for 10 minutes or until the chicken is done.
Begin by starting the chicken: Beat together the egg white, cornstarch, and salt. Stir in the chicken, cover and refrigerate.
Also, prepare the cucumbers and radishes: Sprinkle each with a quarter teaspoon of kosher salt and let sit ten minutes. Squeeze each dry. Stir in one tablespoon of rice wine vinegar and one teaspoon of sugar to each vegetable.
Also, cook the carrots: In a skillet, heat the oil and cook the carrots with the soy sauce and sugar, on medium heat, for a few minutes, just until they're crisp tender. Cover to keep them warm.
Also, continue the Hoisin Chicken: Heat a Dutch Oven or large pot with an inch and a half of the vegetable oil. Heat the oil to 325 degrees. Mix together the stir fry sauce ingredients in a small bowl and set it aside: mirin, soy sauce, sugar, rice vinegar and hoisin. In four batches, fry the chicken just until it turns white. Drain each batch in a colander set over a bowl. In a large skillet, on medium heat, heat two tablespoons of oil from frying the chicken. Add the ginger, scallion, garlic, and red pepper flakes and stir until everything is combined. Add all the chicken to the pan, along with the stir fry sauce. Bring this mixture to a boil. Stir in the cornstarch mixed with water and stir for thirty seconds. Stir in the peanuts.
To serve: Put some rice in four bowls. Top the rice with some chicken and sauce. Also, add the cucumbers, carrots, edamame and radishes. Sprinkle the bowls with the scallions.