Heat oven to 400 F.
In a bowl, toss the peppers and onion with the oil, salt and black pepper. Spread the vegetables out in a single layer on a rimmed sheet pan and roast until they are tender but not caramelized, about 10-15 minutes.
Arrange the sausages on top of the peppers and onion; continue roasting until the peppers are caramelized and the sausages are cooked through, about 8-10 minutes. If the sausages and/or peppers are not browned enough at this point, you can slide the whole pan under the broiler for a minute or two. Watch it carefully so it doesn’t burn.
Meanwhile, in a medium bowl, combine the vinegar and the oregano.
Before serving, toss the peppers and onion with the vinegar mixture; taste and adjust the seasonings if necessary (it might need more vinegar if your onion is very sweet). If you are using the arugula, divide it among individual serving plates; sprinkle with a little olive oil and a few drops of red wine vinegar. Arrange the peppers, onions and sausages over the arugula. Sprinkle with the cheese and serve.