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Preheat oven to 400 °F (200 °C)
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Clean mushrooms and remove the stems. Brush with the olive oil all over, and sprinkle with salt and pepper.
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Arrange mushrooms on a baking sheet, caps up, and bake uncovered for 20 minutes.
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Cut the prosciutto slices into thin strips.
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Mix the ricotta and shredded mozzarella, with the dry oregano - set aside.
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In a separate bowl, mix the whipping cream, with the pesto, and freshly grated nutmeg - set aside.
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Once the mushrooms are baked, pour out the collected liquid from the caps, and distribute the prosciutto at the bottom of each one.
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Top with a heaping spoonful of the cheese mixture.
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Pour the cream and pesto sauce in a baking dish, and arrange the mushrooms on top. Return to the oven for another 10-15 minutes, or until the tops are nicely browned.