In a large bowl, gently combine beef, oats, Worcestershire, ½ teaspoon salt, ¼ teaspoon pepper, garlic powder, onion powder, and milk. Shape into 4 oval patties, about ½-inch thick. Place on a parchment-lined tray, cover loosely, and refrigerate for at least 30 minutes (or overnight).
Melt 1 tablespoon of butter in a large skillet over medium-high heat. Add the patties and cook until browned on both sides, about 3 minutes per side. Transfer to a plate.
Melt the remaining 3 tablespoons of butter in the skillet over medium-high heat. Add onion and cook until softened, about 5 to 7 minutes. Add mushrooms and ½ teaspoon salt; cook for about 5 more minutes. Stir in tomato paste and flour; cook for 2 minutes. Gradually pour in the broth and wine, stirring constantly. Bring to a simmer.
Return the browned patties to the skillet. Cover, reduce heat to medium-low, and cook for 12 to 15 minutes, or until patties are cooked through. Taste and season the gravy with additional salt and pepper if needed.