Gently crumble beef into a large bowl and sprinkle with breadcrumbs. In a separate dish combine egg, Onion Soup Mix, Worcestershire, salt, and pepper and mix to combine.
Combine egg and seasoning mixture with beef and gently mix together with your fingers until well incorporated, then form beef into 4 oval patties.
Place patties in a hot saute pan set over medium heat and cook until a brown crust forms on each side of the patties, and they are almost cooked through, then remove from the pan and set aside.
Place mushrooms in the skillet and saute until they are tender and brown, stirring occasionally. Once tender, remove mushrooms from the pan and set aside.
Turn skillet on low and melt ¼ cup butter in the skillet and sprinkle in flour. Whisk continually to prevent burning while the flour gradually turns a deeper brown. Continue cooking for 5 minutes.
Slowly start adding beef broth a little at a time, while stirring flour to incorporate the liquid. Continue to do this until all broth has been added. Add a few drops of browning sauce at this time to deepen the flavor and color.
Let gravy thicken, while stirring occasionally, then taste for salt and add pepper.
Add beef patties and mushrooms to the gravy and continue cooking until the patties are heated through, then serve.