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Add 1/2 tablespoon of oil to the pan, and stir fry the carrot over medium high heat, sprinkling with a pinch of salt. When the carrot sticks are a half way cooked, about a minute, push them to one side of the pan to make room for the onion slices. Stir-fry the onion slices until translucent, sprinkling with a pinch of salt. Stir-in the scallion and cook briefly. Do not overcook. The vegetables should be crisp. Transfer to the bowl with the noodles.
Separate the egg white and yolk. Lightly beat the white by gently cutting it with a spoon. Stir the yolk with a spoon until smooth. Heat a lightly oiled nonstick skillet over medium low heat. Pour each egg part into a thin layer, by tilting the skillet and/or spreading with a spoon. Cook each side briefly. (Do not brown the egg.) Roll it and thinly slice.