In a large wok or pan over medium heat, heat 1 tablespoon of the oil until it shimmers (this will happen very quickly in a wok). Add the eggs, swirling the pan to spread them around, and cook until set into an omelet, 2 to 3 minutes. Gently turn over and cook on the other side until set, about 2 minutes. Transfer to a cutting board, chop into ½-inch square pieces and set aside.
Season the chicken with ¼ teaspoon of the salt, add to the pan and stir-fry over high heat until cooked through, about 6 minutes. Transfer the chicken to a plate or platter.
Wipe the pan and return it to medium heat. Add the remaining 2 tablespoons of oil, followed by the bell pepper and stir-fry for 1 minute. Add the ginger and garlic and stir-fry until aromatic, a few seconds. Add the rice and stir-fry for 3 minutes or until warmed through.
Return the chicken to the pan, and stir in the soy sauce and oyster sauce. Add the egg and sugar, season with the remaining salt, and stir-fry for another 2 minutes. Add the chopped scallions and sesame oil; stir-fry, mixing everything thoroughly, until warmed through, about 1 minute.
Divide among bowls, top with sliced scallions if desired, and serve hot.
