Bring a medium saucepan filled two-thirds full of water to a boil. Add the farro and cook until tender but still chewy, about 15 minutes. Drain well and cool. (This can be done one day ahead).
Combine the scallion whites, garlic, ginger, crushed red pepper, 1/2 teaspoon salt and sugar in a small bowl and keep near the stove top.
Heat 1 tablespoon of the vegetable oil in a large nonstick skillet or wok over medium-high heat. Add the carrot and cook, stirring constantly, until crisp-tender, about 2 minutes. Add the scallion white mixture and continue to stir fry until fragrant, about 30 seconds. Add the cooled farro, half the scallion greens, the peas, bean sprouts, soy sauce and sesame oil and stir fry until the farro is hot, about 2 minutes. Remove from the heat and transfer to a serving dish.
Wipe out the skillet, add the remaining 1 teaspoon of vegetable oil and place over medium-high heat. Add the eggs and a large pinch of salt; let cook undisturbed until the eggs begin to bubble up, then stir constantly, breaking the eggs up into small pieces, until they are cooked through, about 30 seconds. Scrape eggs over the farro and stir to incorporate. Sprinkle with the remaining scallion green and serve hot.
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