Heat 1 tbsp oil in a large wok or skillet over medium-high heat; add chicken, season with salt and pepper, and stir-fry until cooked through and golden; remove and set aside.
Add remaining 1 tbsp oil; pour in beaten eggs and scramble quickly until just set; break into pieces and transfer to a plate.
In the same wok add onion, garlic, and ginger; stir-fry 1–2 minutes until fragrant.
Add carrots (or mixed peas and carrots) and cook 2–3 minutes until vegetables begin to soften.
Increase heat to high and add cold day-old rice, breaking up any clumps with a spatula.
Drizzle soy sauce and oyster sauce over rice; toss continuously to evenly coat and to allow rice to fry slightly.
Return chicken and scrambled eggs to the wok; fold into the rice until heated through.
Stir in green onions and sesame oil; taste and adjust seasoning with salt, pepper, or extra soy sauce.
If desired, add red pepper flakes or a dash of sriracha for heat and toss to combine.
Serve immediately on warmed plates, garnished with extra sliced green onions.