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.
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.