In a medium bowl, whisk together the eggs and milk and season with salt and pepper.
Melt the butter in a medium non-stick skillet over medium heat. Add the onion and cook, stirring, until golden, about 5 minutes. Add the egg mixture and cook, stirring, until loosely set, about 3 minutes. Sprinkle the cheese over the eggs and stir to combine. Continue to cook, stirring, for 2 to 4 minutes depending on desired firmness of the eggs. Serve immediately.
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