Crumble hamburger into a large bowl and add breadcrumbs, beaten egg, A.1. Sauce, minced garlic, ground mustard, salt, and black pepper. Mix with your hands to thoroughly combine.
Separate meat mixture into six equal balls and flatten into 1" thick patties.
Set 12" skillet over medium-high heat and add butter. Once butter has melted add hamburger steaks and sear on both sides until a brown crust forms. Remove hamburger steaks from the skillet, leaving at least 2 tablespoons of fat in the pan. Don't worry if the meat is not completely cooked through yet.
Reduce the heat to medium and add sliced onions. Saute until tender, approximately 10 minutes.
Reduce heat to medium-low and sprinkle flour over onions. Stir for 1-2 minutes to prevent the flour from burning while the raw flour taste cooks out.
Slowly add beef broth a couple of splashes at a time while continuing to stir to prevent lumps.
Once all liquid has been added, the hamburger steaks can be returned to the skillet. Cook with gravy until the hamburger is cooked through and gravy has thickened.