Break up cold rice with a spoon. Lightly season the diced chicken with a pinch of salt.
Heat 1 tbsp oil in a large skillet or wok over medium-high heat. Scramble eggs just until set; transfer to a plate.
Add remaining oil and the butter. Sear chicken, stirring, until cooked through and lightly browned, 3–4 minutes.
Stir in diced onion, ginger, and garlic. Cook until fragrant and softened, about 1 minute.
Toss in peas and carrots. Stir-fry until hot, 1–2 minutes.
Add rice. Press it into the pan and let it sizzle 1 minute; then toss until the grains separate and steam off.
Pour in soy sauce and oyster sauce. Stir to coat evenly and cook 1 minute more.
Drizzle sesame oil and sprinkle white pepper. Fold in green onions and the scrambled eggs. Adjust salt to taste.
Serve hot straight from the pan.