Finely dice all onions. Mix the minced beef with the egg, mustard, tomato paste, and 2 of the diced onions. Season with oregano, gyros spice, salt, and pepper. Shape into 4 equal oval patties. Press a piece of feta cheese into the center of each patty and enclose with meat.
Heat the clarified butter in a pan over medium-high heat. Fry the bifteki on all sides for 5 min. Transfer to a baking dish.
In the same pan, cook the remaining diced onion for 2 min. Stir in 75 ml water, Metaxa, and cream. Bring to a boil. Season with instant broth to taste. Thicken with sauce thickener if needed.
Pour the sauce over the bifteki. Sprinkle with crumbled feta cheese. Bake at 200°C (392°F) for 15 min.