Prepare the ingredients: Dice onion and carrot into small, uniform pieces. Thaw frozen peas or blanch fresh ones. Beat eggs and set aside.
Heat the pan: Add sesame oil to a large skillet or wok over medium-high heat until shimmering.
Sauté the vegetables: Add onion and carrot, sauté for 3-4 minutes until softened but crisp. Add peas and cook for 1-2 minutes.
Cook the eggs: Push vegetables to one side, pour eggs into the empty side, and scramble gently for 1 minute until just set. Mix with vegetables.
Add the rice: Break up day-old rice clumps and add to the pan. Stir-fry for 4-5 minutes until evenly coated and heated through.
Season the rice: Drizzle soy sauce, mirin, and sake (if using) over rice. Toss well to combine. Season with salt and pepper.
Garnish and serve: Remove from heat, sprinkle with green onions and sesame seeds, and serve hot.