Heat 2 tablespoons of the oil in a skillet over medium heat. Add the mushroom caps and cook until brown and tender, about 8 minutes. Stir the remaining olive oil, vinegar, onion, water, garlic, coriander, pepper flakes, salt, and pepper together in a bowl. Add the oil mixture to the mushrooms and cook until tender and marinade has reduced, about 10 to 12 minutes.
Transfer to a bowl and cool. Serve now or refrigerate for up to a week.
Meanwhile, put the almonds in a small dry skillet over medium heat, toss until evenly toasted and fragrant, about 2 minutes. Set aside.
Toss the arugula with the marinated mushrooms and marinade in a large bowl and divide among 4 plates. Shave the cheese and scatter the almonds over the top.
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