Mix the ground beef, egg, breadcrumbs, white or black pepper, salt, grated onion, and minced garlic gently with your hands in a large bowl.
Divide the mixture into 4-6 equally sized portions, then gently shape them into patties and set aside.
In a medium bowl, whisk the flour into the broth and water, and set aside.
Heat a lightly oiled pan over medium-high heat. Sear patties for 2-3 minutes on each side or until browned. Remove them from the pan and set aside.
Add 2 tablespoons of butter, followed by mushrooms, onions, thyme, garlic, Worcestershire sauce, and Creole seasoning, and sauté for about 6 minutes or until the mushrooms and onions are tender.
Add the broth and flour mixture, and bring to a boil. Return the patties and the drippings to the skillet, and adjust seasonings with salt and pepper as needed.
Cover, reduce the heat to low, and simmer for 8-10 minutes. Adjust the consistency with more stock or water, and serve immediately with your chosen sides.