Salt and pepper the beef to your liking. Mix together ground beef, egg, breadcrumbs, parmassan, and garlic in a medium bowl until combined. Shape the ground beef into 4 elongated patties.
Heat a large, cast iron skillet over high heat. Add the oil and let heat until it begins to shimmer. Add the patties and let sear for 3-4 min. Flip and sear the other side, approx. 6-8 min total. Patties should be cooked to approx. 145º internal temp at this point. If your patties are thicker, you may have to cook them longer.
Remove the patties from the pan and transfer to a place or shallow bowl. Reduce the heat to medium and add the sliced mushrooms and onions to the pan, stirring occasionally until soft, about 4-5 min. Stir in the tomato paste, Worcestershire sauce, and beef broth. Simmer for approx 5 min until it reduces, and stir in half-and-half. Continue to reduce for another 3-4 min.
Add the patties and any juices back to the pan, and continue to simmer until internal temperature reaches at least 160º. Spoon the sauve over the patties as they continue to cook. [NOTE: You can cook them less, but there's a danger to eating undercooked ground beef, so it is not recommended. We found the sauce kept them nice and moist, despite cooking them to safe temp]
Transfer the patties to a large place and spoon the sauce, mushroom, and onions over the steak. Garnish with parsley or chives, and serve while hot.