Heat broiler. Toast bagel halves cut-side up on a sheet 1 to 2 minutes until just golden, watching closely; set aside.
Whisk eggs, milk, scallions, salt, and pepper. Divide cheese into four equal portions.
Melt 1/2 tablespoon butter in an 8-inch nonstick skillet over medium-low. Add one-quarter egg mixture; tilt into a thin, even layer.
Cook about 2 minutes until mostly set. Add one cheese portion in center; fold four sides inward into a square. Carefully turn; cook 1 to 2 minutes until egg reaches 160°F and cheese melts.
Slide onto one bagel bottom. Repeat with remaining butter, eggs, and cheese for four bagels.
Close with toasted tops; rest 1 minute. Halve with a serrated knife without flattening eggs.