Soak the shiitakes in hot water for at least 20 minutes.
Get your pasta water boiling.
Cut onions in half lengthwise, then in thin slices. Heat olive oil in a large frying pan until it shimmers. Add onions and sauté on medium heat until they are well caramelized. As onions start to stick to the pan, add little bits of the Madeira and scrape them loose. The Madeira will evaporate and also enhance the caramelization. When onions are about 2/3 done, put the linguini in the water pot to cook.
When the onions are finished, drain and squeeze excess moisture from mushrooms and add them to the pan.
As soon as the linguini is cooked, scoop it out of the water and put it straight into the pan. Add the arugula and gently toss everything, allowing the heat and moisture from the hot pasta to wilt the arugula.
Serve garnished with a sprinkling of balsamic vinegar, salt and pepper.