In a bowl, combine breadcrumbs and milk; let sit for 1-2 minutes. Mix in ground beef, egg, chopped onion, garlic, Worcestershire sauce, mustard, thyme, rosemary, salt, and pepper until just combined.
Shape into four oval patties and chill in the fridge for at least 30 minutes.
Heat olive oil and butter in a skillet over medium-high heat. Sear patties for about 3-4 minutes on each side until browned.
Remove patties and sauté onions and mushrooms in the same skillet until softened. Add garlic and cook briefly.
Sprinkle flour over the mixture; stir for 1-2 minutes before gradually adding beef broth while stirring continuously.
Return patties to the skillet; let simmer covered for about 15-20 minutes.