Mix. Combine the turkey, garlic, salt, pepper, ketchup, Worcestershire, steak sauce, egg, and panko in a large bowl. Mix gently until just combined. Shape. Divide into 4 portions and shape each into an oval patty about 3/4 inch thick.
Brown. Heat the oil in a large nonstick skillet over medium high heat. Cook the patties 2 to 3 minutes per side until deeply browned, then move them to a plate. Cook the vegetables. Pour off all but about 1 tbsp oil and reduce the heat to medium low. Melt the butter, add the onions and mushrooms, and cook 4 to 5 minutes until softened.
Add flour. Stir in the flour and cook 1 to 2 minutes until no dry patches remain.
Make the gravy. Raise the heat to medium, pour in the broth, Worcestershire, salt, and pepper, and scrape up the browned bits. Bring to a simmer.
Simmer. Return the patties to the skillet, cover, lower the heat, and cook 20 minutes, turning at least twice.
Finish. Uncover for the last 5 minutes to thicken the gravy. Check that the patties reach 165 degrees F.