Heat a large cast iron skillet or wok over high heat until hot, about 2 minutes. Add 1 tablespoon oil.
In a bowl, beat egg with 2 teaspoons water and a pinch salt; add to the skillet. Cook on medium-low, stirring to form large soft curds, about 30 seconds. Remove and transfer to a plate.
Return skillet to high heat and add 2 tablespoons oil, the carrots, and whites of the green onions. Cook until lightly golden, about 2 minutes. Add garlic and ginger and cook, stirring, until fragrant, 1 minute.
Add rice, peas, and cooked eggs to skillet. Pour in soy sauce and cook, stirring until heated through, 1 minute. Season with salt and pepper and stir in the remaining green onions.