In a large, deep skillet with a tight fitting lid, get the olive oil heating over medium heat. In a large bowl, combine the ground beef, egg, breadcrumbs, onions, garlic, Worcestershire, ketchup, salt and pepper until well mixed. Then use your hands to form them into 4 patties. Place the patties in the skillet and cook them for 5-7 minutes per side, until a nicely browned crust forms. Once done with that, set them on a plate and cover loosely with foil.
Wipe out the skillet and add the butter. Once it melts, add the onions. Cook until they begin to soften, then add the mushrooms and ground thyme. Cook another 4-7 minutes, until the onions are translucent and the mushrooms are browned and tender. Add the garlic and cook until just fragrant, then sprinkle over the flour and stir to combine. Cook another 2 or so minutes, stirring it around often, before stirring in the Worcestershire, ketchup and dijon mustard.
Stir the broth into the onion mixture and season with the salt and pepper. Let this come up to a simmer. Then nestle the patties back into the skillet, spooning some of the gravy over the tops. Cover with the lid and cook for 10 or so minutes, until the patties are fully cooked through. Then serve.
