Place potatoes, salt and enough water to cover the cut potatoes, in a medium pot. Bring to a boil over high heat.
Once boiling, reduce heat and simmer until potatoes are tender when pierced with a fork. Approximately 10 minutes.
Drain potatoes and place in a single layer on a large baking sheet to allow the potatoes to cool completely and quickly, about 15 minutes.
Meanwhile, cook the sliced sausage in a large, non-stick skillet over medium-high heat, stirring often, until cooked through, about 10 minutes.
Remove with a slotted spoon and drain on paper towels if necessary. Leave residual drippings in the skillet.
After potatoes have cooled, place potatoes in the hot drippings, stirring occasionally until potatoes are brown and crisp, 10-15 minutes.
Add onions to the pan and cook, stirring occasionally until tender, about 10 minutes.
Add vinegar and cook an additional 40 seconds.
Stir in sausage and arugula and cook. Stir until arugula is wilted, about 5-7 minutes.
Transfer to a platter, top with cheese and serve.