Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until almost tender but still firm, about 15 minutes. Drain and finely chop.
Cook beef in a large cast-iron or non-stick skillet over medium heat until fat begins to render. Stir green onions, salt, and pepper into beef. Add garlic and butter; cook and stir until butter is melted and garlic is fragrant, 1 to 2 minutes.
Stir potatoes, salsa, and a pinch of salt into beef mixture. Pat mixture down in the skillet with a spatula and divide hash into 4 sections. Cook, without stirring, until a crust forms on the bottom, about 5 minutes. Flip sections and cook until a crust forms on the bottom again, about 5 minutes more. Repeat cooking and flipping process, flipping every 5 minutes, until hash stays together and has a golden crust, 10 to 15 minutes.