Preheat oven to 400°. Cut puff pastry sheet into four equal pieces, approx. 4” by 5”. Place on parchment-lined sheet pan and brush lightly with beaten egg. Bake until golden and puffed high in center; check to ensure bottom is also browned. Remove from heat and cool to handle.
Meanwhile, sauté shallots in 2 T butter and 1 T oil until soft, 3 min. Add garlic, salt and pepper and cook 1 min more. Add remaining butter, oil and mushrooms and cook 3-4 min. Add herbs, then deglaze pan with sherry. Add cream and reduce gently until thickened.
Remove from heat and stir in Parmesan, then taste for seasoning. The mushrooms should be tender yet still colorful and individually recognizable. Keep warm.
With a sharp knife, cut along top edges of baked pastry leaving a border, going through top layers of puff pastry to create a box; do not cut edges or through the bottom. Press layers down in center to create a cavity. Spoon mushroom mixture into each tart, filling with ½ c or so of mushroom mixture. Top with a sprinkle of cheese, as desired. Return to oven and bake 5-7 min until cheese has melted and tops are golden brown. Serve immediately with fresh herbs sprinkled over each.