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Preheat your oven to 400°.
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Snap off the woody ends of the asparagus. If you have thick asparagus cut in half lengthwise. Cut into 2" pieces. Toss the Asparagus with the olive oil and saute about 5 minutes in a skillet, set aside.
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In same sauté pan, heal until it is hot and then add the butter and chopped mushrooms. Saute these for about 5-7 minutes. Add the fresh thyme leaves, salt and pepper. Cook these until they just start to get tender, about 10 minutes.Add the garlic and cook another 5 minutes. Remove from heat.
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Unfold the thawed puffed pastry sheets and cut each sheet into 9 squares. Using a small knife, score 1/2" border (do not cut through pastry) around inside edges of each square. Bake these for about 5 minutes until they start to rise. Remove from oven Spread a small about (about 1 Tbsp.) of the Mascarpone cheese in the center of the square. Sprinkle about 1-2 Tbsp. of the Fontina cheese on top of the mascarpone cheese. Bake these for about 5-7 minutes and then remove from oven and top with some of the asparagus pieces and sautéed mushrooms. Sprinkle with Parmesan cheese and bake for another 10-15 minutes until golden brown. I think I cooked mine quite a bit longer to the the brownest/crunchiness I wanted.
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Serve and enjoy.