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Place a foil-lined baking sheet on a middle rack and preheat oven to 425 degrees. Bring a large pot of salted water to a boil. Wash and dry all produce. Quarter and peel onion; finely dice three quarters and thinly slice remaining. Mince garlic. Finely chop feta. Quarter mushrooms. Peel and trim carrot, then finely chop. Thinly slice cucumber into rounds.