Combine the beef, egg, feta, garlic, salt, and pepper. Mix until the ingredients are well blended. Make 12 to 14 equal hamburgers from the mixture.
You'll need a 12-inch or larger skillet to prevent the sauce from bubbling over. Heat the skillet over medium-high heat, and cook the patties until done. Depending on the size of your skillet, you may have to cook multiple batches.
After they're cooked, remove the patties from the skillet. You need 4 tablespoons of grease left in the pan. Drain extra grease out of the pan.
For the sauce
You need about 4 tablespoons of fat in the pan. If there's not enough from cooking the meat, you can use butter. Add butter, beef broth, heavy cream, fresh herbs, steak sauce, garlic, salt, pepper, and crushed red pepper flakes to the skillet. Heat it to medium-high.
Bring mixture to a boil while whisking constantly then reduce the heat to low or simmer. Simmer for 10 to 12 minutes then turn it off while you cook the hamburger.
Add the hamburger steak back into the pan and spoon sauce over the top. Simmer for 5 minutes then serve.
Notes
*Or 1 teaspoon dried herbs
Salt, crushed black pepper, and crushed red pepper flakes for added flavor.