Slice oyster mushroom to create large, thin slices. Toss with some olive oil, salt and pepper and lay out on a metal baking sheet so that the slices are not overlapping. Roast in oven until brown and crisp, turning and tossing the mushrooms occasionally, about 20 minutes.
Combine vinegar and olive oil in a bowl and season to taste. Add arugula, mushrooms and any accumulated oil or juices from roasting pan. Divide among 4 plates and scatter cheese on top.