3/4 pound shiitake mushrooms, stemmed and large ones halved
1 tablespoon chopped fresh tarragon leaves, plus more for serving
Coarse salt and ground pepper
6 large eggs
Grated Parmesan, for serving
On a lightly floured piece of parchment paper, roll out puff pastry to an 11-by-14-inch rectangle. Trim edges to make a neat rectangle and transfer pastry on parchment to a rimmed baking sheet (trim parchment to fit, if necessary). With a sharp knife, score a 1/2-inch border around edges. With a fork, prick pastry all over inside border. Refrigerate 20 minutes.
Preheat oven to 375 degrees. In a large skillet, melt butter over medium. Add onion and cook, stirring occasionally, until soft and golden, 10 to 15 minutes. Add mushrooms and tarragon and cook until mushrooms begin to soften, about 8 minutes; season with salt and pepper.
Bake puff pastry until light golden, about 15 minutes. Remove from oven and top with mushroom mixture, staying within border. Carefully crack eggs on top of mushroom mixture, spacing them evenly, and bake until pastry is golden and egg whites are set but yolks are still runny, about 10 minutes. To serve, sprinkle with tarragon and Parmesan and cut into 6 squares.
Perfect Pastry: Docking, or pricking the dough, allows steam to escape, thus preventing the inner part from rising as much as the edges.