Stir together the 2 tbsp mayonnaise, 1 tsp Dijon mustard, 1 tsp yellow mustard, and ½ tsp white wine vinegar until smooth. Refrigerate while you cook.

Heat a skillet over medium-high. Cook the ¼ cup sliced onion in the dry pan for 2 minutes, then push aside. Press the 3 oz ground beef into a thin patty, season with ¼ tsp salt and ¼ tsp black pepper, and sear for 3 minutes per side until deeply browned. Remove the beef and onion and set aside.

Reduce heat to medium. Melt 1 tbsp butter, crack both eggs into the pan, immediately break the yolks, and season with the remaining ¼ tsp salt. Let the egg set for 1 minute, then fold it into thirds. Lay the American cheese slice on top in the last 30 seconds until melted, 2 to 3 minutes total.

Add the remaining 1 tbsp butter to the pan over medium heat. Place the bagel halves cut-side down and toast for 2 minutes until golden.

Spread the sauce on both bagel halves, layer on the beef patty, onions, and cheesy folded egg, and serve immediately cut-side up.