Whisk eggs, milk, and salt until streak-free.

Melt butter in a nonstick skillet over medium-low. Cook eggs in a thin sheet; fold into 4 bread-size rectangles when softly set.

Top each of 4 bread slices with ham, an egg rectangle, American cheese, Swiss, and another bread slice.

Spread outsides with mayonnaise. Cook in a clean skillet over medium-low, 3 to 4 minutes per side.

Remove when both sides are evenly brown and cheese softens through the egg layer.