In large pot of boiling salted water, cook pasta until al dente. Reserving 1/2 cup/125 mL of the water, drain.
Meanwhile, in large skillet, heat oil over medium-high heat; fry shallots and garlic until garlic just begins to turn light golden, about 2 minutes. Add mushrooms and sauté until tender, about 2 minutes. Add vermouth and salt; cook until liquid is reduced by half.
Add reserved water, pasta, arugula and butter; cook until arugula is wilted, butter is melted and liquid no longer pools in bottom of pan. Serve with cheese on side.
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