Heat oven to 425 degrees. In a medium bowl, use a fork or a wooden spoon to mash together the goat cheese, egg, garlic, tarragon, lemon zest, salt and nutmeg until smooth. Switch to a whisk and beat in the crème fraîche until smooth.
On a lightly floured surface, roll out puff pastry into a 13-by-11-inch rectangle about ⅛-inch thick. Transfer the dough to a parchment-lined cookie sheet. With a sharp knife, lightly score a ½-inch border around the edges of the puff pastry.
Spread the crème fraîche mixture evenly inside the scored border. Line up the asparagus spears on top, and brush them with olive oil. Sprinkle some salt and the grated Parmesan over the asparagus.
Bake until the pastry is puffed and golden, 25 to 30 minutes. Let it cool on the cookie sheet for at least 15 minutes or up to 4 hours before serving. Then sprinkle black pepper, red-pepper flakes (if using), the shaved Parmesan and tarragon leaves. Drizzle a little oil on top.