To marinate the chicken, in a bowl, combine the chicken breast with salt and mix until absorbed. Add the white pepper, soy sauce, and cornstarch slurry and mix until evenly coated. Add cooking oil and combine.
To make the teriyaki sauce, in a bowl, combine the soy sauce, sake, mirin, and sugar and stir until the sugar dissolves.
To stir-fry the chicken, in a wok over high heat, add 1 tablespoon cooking oil. Add the marinated chicken and stir-fry until fully cooked, about 3 minutes. Add the garlic and stir-fry briefly until fragrant. Remove from the wok.
To stir-fry the vegetables, in the same wok over high heat, add 1 tablespoon cooking oil. Add the onion and carrot and stir-fry for about 30 seconds until aromatic. Remove.
To make the fried rice, in the wok over high heat, add the remaining cooking oil. Add the eggs and scramble until partially set. Add the rice and stir-fry for about 2 minutes until the grains separate. Return the chicken and vegetables to the wok and pour in the teriyaki sauce. Toss everything together and stir in the scallions.