Preheat the skillet over medium-high heat and add 1 tablespoon of oil. Once shimmering, form the ground beef into four equal-sized patties and place them in the skillet. Cook until golden brown on both sides, about 4-5 minutes per side, then remove and set aside.
Reduce heat to medium. Add the sliced onions to the same skillet and cook, stirring occasionally, until they are soft and caramelized, about 8-10 minutes. The onions should turn a deep golden brown and release a sweet aroma.
Add the Worcestershire sauce, mustard, and beef broth to the skillet with the onions. Stir well to combine and scrape up any browned bits from the bottom of the pan. Bring the mixture to a gentle simmer and cook until it slightly thickens, about 3-5 minutes.
Return the browned beef patties to the skillet, spooning some of the gravy and onions over each. Reduce heat to low, cover loosely, and cook for an additional 5 minutes until the patties are heated through and coated with a glossy gravy.
Remove from heat and plate the Salisbury steaks, spooning the onions and gravy generously over the top. Serve hot, allowing the glossy sauce to coat the tender meat.
