1 17.3-ounce package frozen puff pastry (2 sheets), thawed
1/4 cup (1/2 stick) unsalted butter
12 ounces fresh shiitake mushrooms, stemmed, caps cut into 1/4-inch-wide strips
1 teaspoon coarse kosher salt, divided
1/2 teaspoon coarsely ground black pepper, divided
1 pound slender asparagus spears, trimmed, cut on diagonal into 1-inch pieces
1 1/2 teaspoons chopped fresh thyme
1 1/2 teaspoons finely grated lemon peel
1/2 cup sour cream
1/2 cup (packed) coarsely grated Gruyère cheese (about 2 ounces)
Grating or two of fresh nutmeg
Fresh thyme sprigs (for garnish)
Preheat to 400°F.
Roll out each pastry sheet on work surface to 10-inch square. Cut each into 4 squares. Using small knife, score 1/2-inch border (do not cut through pastry) around inside edges of each square. Arrange squares on 2 rimmed baking sheets. DO AHEAD: Can be made 1 day ahead. Cover and chill.
Melt butter in heavy large skillet over medium-high heat. Add mushrooms; sprinkle with 1/4 teaspoon coarse salt and 1/4 teaspoon pepper. Sauté until tender and lightly browned, about 4 minutes. Transfer mushrooms to large bowl; cool 15 minutes. Add asparagus, chopped thyme, lemon peel, 3/4 teaspoon coarse salt, and 1/4 teaspoon pepper to mushrooms. Mix in sour cream and cheese. Mound filling atop pastry squares, leaving 1/2-inch plain border.
Bake tarts 12 minutes. Reverse sheets. Continue to bake tarts until crusts are puffed and golden and filling is cooked through, about 10 minutes longer. Serve.