Prepare the sauce: In a small bowl whisk together mayonnaise, Dijon mustard, lemon juice, honey, and cayenne. Cover and refrigerate until serving.
Toast the bagels: Spread butter evenly on cut sides. Place bagels on a baking sheet, cut-side up, and broil 1–2 minutes until golden. Watch closely to prevent burning.
Whisk eggs and milk with a pinch of salt and pepper. Heat a nonstick skillet over medium-low, coat with a little butter, pour in the egg mixture, and let it set slightly. Fold edges inward as it firms to create a thick, round folded egg—about 2–3 minutes.
Assemble sandwiches: Place one cheese slice on each bottom bagel half, top with folded egg, choice of bacon or steak, another cheese slice, and a generous smear of breakfast sauce. Cap with the top half of the bagel.
Serve immediately, wrapping in parchment for a classic diner feel.