In a medium bowl, mix together milk and potato flakes. Stir in 1 tablespoon on onion soup mix.
Add ground beef and 1/4 teaspoon pepper. Mix together and form into 4 oval-shaped patties. Place on a plate and cover with plastic wrap. Refrigerate at least 30 minutes.
Melt 1 tablespoon butter in a large nonstick pan over medium-high heat. Add patties and cook until nicely browned on both sides, about 5 minutes per side. Transfer to a plate and set aside.
Add remaining 3 tablespoons butter and 1 tablespoon vegetable oil to pan and place over medium heat. Add sliced onion.
Once onion begins to soften, after about 2 minutes, add mushrooms and cook until mushrooms are browned and most of liquid has evaporated.
Sprinkle flour on top of onion/mushroom mixture. Stir and cook for 1 minute.
Stir in remaining onion soup mix and ketchup.
Gradually whisk in broth. Bring to a simmer.
Return patties to pan, cover, reduce heat to low, and cook 12 to 14 minutes.
Check for seasoning and add salt and pepper to taste.
Serve over mashed potatoes.