Place the peeled and quartered potatoes in a large pot and cover with cold water by 1 inch. Add a pinch of salt. Bring to a boil over high heat, then reduce to a simmer and cook for 15-20 minutes, or until the potatoes are fork-tender. Drain well. Return the potatoes to the hot pot and let them steam dry for 1 minute. Add the warm milk, butter, sour cream, salt, and white pepper. Mash until smooth and creamy. Cover and keep warm.
In a large bowl, combine the ground beef, breadcrumbs, beaten egg, Worcestershire sauce, garlic powder, onion powder, salt, and pepper. Mix gently with your hands until just combined—do not overmix. Divide the mixture into 4 equal portions and shape each into a 3/4-inch thick oval patty.
Heat the vegetable oil in a large skillet (preferably cast iron or stainless steel) over medium-high heat. Once hot, add the patties. Cook for 4-5 minutes per side, until a deep brown crust forms. Remove the patties to a plate; they will not be fully cooked through. Reduce heat to medium. Add the sliced onions to the same skillet with the beef drippings. Cook, stirring occasionally, for 5-7 minutes until softened and golden brown.
Push the onions to the side of the skillet. Add the 2 tablespoons of butter to the center and let it melt. Sprinkle the flour over the butter and cook, stirring constantly, for 1 minute to make a roux. Gradually whisk in the beef broth, ensuring no lumps form. Add the Worcestershire sauce, Dijon mustard, and black pepper. Bring to a simmer, stirring constantly, until the gravy thickens, about 3-5 minutes. Taste and adjust salt if needed.
Return the seared beef patties and any accumulated juices to the skillet, nestling them into the gravy. Reduce heat to low, cover, and simmer gently for 10-12 minutes, until the patties are cooked through and the gravy has thickened further.
While the patties simmer, steam or boil the green beans until crisp-tender, about 5-7 minutes. Drain and toss with butter, garlic powder, salt, and pepper.
Spoon a generous portion of mashed potatoes onto each plate. Place a Salisbury steak patty on top of the potatoes. Ladle plenty of onion gravy over the patty and potatoes. Serve immediately with the buttered green beans on the side.