Heat 2 tablespoons of butter in a large skillet over medium heat. Add sliced white onions and sauté until translucent, about 5 minutes.
Add ground chuck, Worcestershire sauce, salt, garlic powder, and black pepper. Break up the beef and cook until browned and cooked through, about 8-10 minutes. Remove from heat and set aside.
Whisk the eggs in a medium bowl until smooth.
Melt 1 tablespoon of butter in a clean skillet over medium-low heat. Pour in whisked eggs and gently scramble until just set but still soft and creamy. Remove from heat and keep warm.
In a small bowl, combine mayonnaise, yellow mustard, and dried dill. Mix well.
Spread remaining 2 tablespoons butter on cut sides of bagels. Toast in toaster oven or under broiler until golden and crispy, about 2-3 minutes.
Spread dill mustard mayo on bottom half of each toasted bagel. Add cooked steak and onion mixture, then scrambled eggs. Top with a slice of American cheese while warm. Close with top half of bagel.
Serve immediately while hot and melty. Enjoy your Homemade McDonalds Steak Egg and Cheese Bagel!