Preheat oven to 450F/230C. Lightly spray a non-stick rimmed baking sheet with non-stick spray.
Place potatoes in a medium bowl and drizzle the 1 tablespoon of melted butter over them; toss to coat, then add the salt and pepper and toss again.
Arrange potatoes, cut side down, on the prepared baking sheet. Bake for 30 minutes or until potatoes are done and nicely browned on the cut side.
While the potatoes are baking, mix the ground beef, panko crumbs and onion soup mix in a large bowl. Divide into 4 equal portions and shape into burgers about 1/2-inch/1.25 cm thick.
Melt 1 tablespoon of butter in a large skillet over medium heat. Add the patties to the pan and cook for about 4 minutes each side. Patties should have an internal temperature of 130F/55C. See NOTES
Remove patties from the pan and tent with foil to keep warm.
In the same pan, increase the heat to medium-high, add the mushrooms. Cook until well seared. Stir in the flour and cook for 1 minute. Slowly add the beef stock whisking constantly to avoid lumps. Remove from heat and add the final 2 tablespoons of butter; stir to combine.
Pour the sauce over the patties and serve with the potatoes.
If you want the patties more done, place them on a baking dish and put them in the oven with the potatoes for about 5 minutes while making the sauce.