Beat eggs with a pinch of salt.
Heat 1 teaspoon oil in a wok or large skillet over medium-high. Cook eggs, stirring, until just set, 30 to 60 seconds. Transfer to a plate.
Add ½ teaspoon oil to the wok. Sauté carrots with a pinch of salt until starting to brown, 2 to 3 minutes.
Add ham and cook until heated through, 1 to 2 minutes. Transfer to the plate with eggs.
Add scallion whites and remaining 1½ teaspoons oil to the wok. Cook until fragrant, about 1 minute.
Stir in rice, breaking up clumps. Spread in an even layer and season with black pepper and remaining salt.
Cook, stirring and pressing, until rice is heated through, 2 to 4 minutes.
Add peas, egg mixture, and scallion greens. Stir and break up eggs.
Cook until peas are warmed through, about 2 minutes. Serve immediately.