Heat the broiler and place bagel halves cut side up on a sheet pan. Toast for 1 to 2 minutes until just golden on the cut faces, watching closely, then set aside.
Whisk eggs, milk, scallions, salt and pepper in a bowl. Divide cheese into four equal portions.
Melt 1/2 tablespoon butter in an 8-inch nonstick skillet over medium-low heat. Pour in one-quarter of the egg mixture and tilt the pan to make a thin even layer.
Cook for about 2 minutes until mostly set, put one cheese portion in the center and fold the four sides of the egg inward to make a square. Turn carefully and cook for 1 to 2 minutes until the egg reaches 160°F and the cheese melts.
Slide the folded egg onto one bagel bottom. Repeat with the remaining butter, egg mixture and cheese to make four filled bagels.
Close with the toasted tops, rest for 1 minute and cut in half with a serrated knife without pressing the egg flat.