Place 1 cheese slice on cut side of each bagel half. Combine egg, salt, and pepper in a bowl, stirring with a whisk. Melt butter in a small nonstick skillet over medium heat. Add egg mixture; cook, without stirring, until mixture sets on bottom. Draw a spatula across bottom of pan to form curds. Continue cooking, stirring occasionally, until egg mixture is thickened but still moist. Immediately spoon half of egg over cheese on each bagel half.
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