Make this hash even heartier by adding a fried egg on top.
Place potatoes, salt, and water to cover in a medium saucepan. Bring to a boil over high; reduce heat to medium-low, and simmer until tender when pierced, 10 minutes. Drain, and place in a single layer on a baking sheet. Cool completely, 15 minutes.
Meanwhile, cook the sausage in a large skillet over medium-high, stirring often, until browned, 10 minutes. Remove with a slotted spoon, and drain on paper towels, reserving drippings in skillet.
Cook potatoes in hot drippings over medium, stirring occasionally, until potatoes are brown and crisp, 10 minutes. Add onions, and cook, stirring occasionally, until tender, 10 minutes. Add vinegar, and cook 30 seconds.
Stir in sausage and arugula, and cook, stirring until arugula is wilted, about 5 minutes. Transfer to a serving platter. Top with Parmesan.