In a heavy large skillet, heat 2 TB olive oil over medium high heat. Add onions and stir 2-3 minutes. Add garlic and stir 1 minute. Reserve half of this mixture for later.
Add the other half of the garlic/onion mixture to a large bowl with ground beef. Add breadcrumbs, beaten egg, mustard, ketchup, salt, Worcestershire sauce, and pepper to the bowl. Stir to combine mixture well. Form beef mixture into 7 even sized patties, about ½” thick each.
In heavy large skillet, add another TB of oil and heat over medium high heat. Cook beef patties 3 minutes per side or until nicely browned on both sides and fully cooked through — do this in 2 batches, adding olive oil as needed.
While patties are cooking, make the gravy: In a separate saucepan, add the butter over medium high heat. Add flour to melted butter and whisk until mixture is a thick roux. Add broth, mustard, Worcestershire, and sugar to the saucepan; constantly whisk until mixture is smooth. Bring to a boil and let sauce boil about 5 minutes or until thickened. Add reserved onion/garlic mixture from earlier, plus mushrooms, stirring to combine well. Add salt and pepper to taste, if needed.
Pour finished gravy over beef patties in original skillet and heat just to warm through. Serve immediately.