Gently mix ground beef, onion, egg, milk, breadcrumbs, Worcestershire, Dijon, salt, and pepper, then shape into four oval patties.
Heat olive oil in a skillet and brown patties for about 4 minutes per side, then remove and set aside.
Add more oil to the skillet and cook mushrooms for about 5 minutes until softened and lightly golden.
Stir in garlic, cook briefly, then pour in beef broth and scrape up the browned bits.
Simmer and whisk in the cornstarch slurry, cooking until the gravy thickens.
Reduce heat, stir in butter, then return patties to the skillet and spoon gravy over them.
Cover and simmer for about 10 minutes until patties are cooked through and the gravy is smooth.