Preheat oven to 400°F. Prick potatoes with a fork and bake for about 30 minutes, until fork tender. Peel if desired, and cut into 1/2-inch dice.
Heat a heavy 9- or 10-inch skillet over medium heat until hot. Add beef and break up with a wooden spoon. Season with 1/4 tsp. salt. Cook until browned, about 6 minutes; transfer to a plate.
Add 1 Tbsp. oil to skillet. Add onion and cook, stirring, until softened, about 4 minutes. Add potatoes, 1 Tbsp. butter and remaining oil. Season with 1/2 tsp. salt and 1/4 tsp. pepper and cook, turning occasionally, until potatoes are golden and crisp, about 5 minutes. Return cooked ground beef to pan and cook until heated through.
Melt 1 Tbsp. butter in a nonstick skillet. Crack 4 eggs into pan, season with 1/8 tsp. salt and pepper, and fry. Repeat with remaining butter, eggs, salt and pepper. Serve 2 eggs with each portion of hash.