Put 1 tablespoon oil in a wok or a large skillet, preferably nonstick, and turn heat to high. A minute later, add onion and pepper and cook, stirring occasionally, until softened and beginning to brown, 5 to 10 minutes. With slotted spoon, remove to a bowl.
Add chicken to skillet and cook over high heat, stirring infrequently, until nicely browned, 5 to 10 minutes. Add to bowl with vegetables. Drain peas if necessary and add them to skillet; cook, shaking pan, for about a minute, or until hot. Add them to bowl.
Put remaining oil in skillet, followed by garlic and ginger. About 15 seconds later, add rice, a bit at a time, breaking up clumps with your fingers and tossing it with oil. When all rice is added, make a well in the center and break eggs into it; scramble them a bit, then incorporate into rice.
Return chicken and vegetables to skillet and stir to integrate. Add wine or water and cook, stirring, for about a minute. Add soy sauce and sesame oil, then taste and add salt and pepper if necessary. Turn off heat, stir in scallion or cilantro, and serve.