In a bowl, gently mix together the beef, bread crumbs, half of the onion, garlic, egg, salt, and pepper; form into 4 patties, each ¾ inch thick.
Heat the oil in a large skillet over med-high heat; cook the burgers until just cooked through, about 3-5 minutes per side.
Remove patties from the pan and set aside; reserve pan drippings.
In a bowl, mix the cornstarch with 1 cup broth.
To the reserved drippings, add the remaining onion and mushrooms; cook, stirring over med-high heat for about 2 minutes or until just softened, scraping the browned bits off the bottom of the pan.
Add in the remaining 1 cup broth, ketchup, Worcestershire sauce, and mustard; whisk in the broth/cornstarch mixture and bring to a boil for 1 minute; decrease heat to low, return burgers to the pan, and simmer for 3-5 minutes.
Serve on hamburger buns; top off with some of the pan juices.