In a frying pan, melt the butter over low heat, swirling around the pan to evenly coat. Add the onion and cook, stirring with a wooden spoon, until caramelized, about 20 minutes.

In a small bowl, beat the eggs together with a fork. Season the eggs with salt and pepper. Distribute the cut pieces of lox evenly in the pan. Pour in the egg and allow it to set for a few seconds. With a wooden spoon, pull the edges of the eggs in toward the center of the pan until the eggs begin to curd, about 30 seconds. Continue to stir until the eggs have just set, but still have a glossy finish about 1 minute. Blanket the eggs with slices of American cheese and cover the pan until the cheese has melted, about 1 minute. Serve with a toasted bagel spread with butter or cream cheese.