Whisk soy sauce, oyster sauce, rice vinegar, sesame oil, and white pepper in a small bowl; set aside.
Break up cold cooked rice with a fork so the grains separate easily.
Heat 1 tsp oil in a large skillet or wok over medium. Pour in beaten eggs, scramble just set, then transfer to a plate.
Increase heat to medium-high. Add 2 tsp oil, then the shrimp. Stir-fry until pink and opaque, 2–3 minutes. Move to the plate with eggs.
Add remaining oil and the butter. Sauté yellow onion for 2 minutes, then stir in garlic and ginger for 30 seconds.
Toss in peas and carrots and cook until warmed through, about 1 minute.
Add rice. Spread it out and let it sizzle undisturbed 1 minute, then stir-fry until hot and lightly crisp.
Return shrimp and eggs to the pan. Pour in the sauce and stir until everything is evenly coated.
Fold in most of the green onions. Season with salt and red pepper flakes. Garnish with remaining green onions and serve.