Step 1

Preheat the oven to 375°. Spread the squash on a baking sheet and toss with 2 tablespoons of the oil. Season with salt and pepper. Bake for 25 minutes, until just tender. Transfer to a bowl. Increase the temperature to 400°.

Step 2

Meanwhile, in a skillet, melt the butter in the remaining 2 tablespoons of oil. Add the shallot and cook over moderate heat, stirring, until softened. Add the mushrooms, cover and cook, stirring, until tender, about 7 minutes. Season with salt and pepper and remove from the heat. Toss with the vinegar and nutmeg and add to the squash.

Step 3

Line a baking sheet with parchment. On a floured work surface, roll the puff pastry out to 12 1/2 by 14 1/2 inches. Using a knife, trim the pastry to 12 by 14 inches. Transfer to the baking sheet and prick the pastry with a fork all over except for a 1/2-inch border. Bake the pastry for 20 minutes, until golden; pierce with a fork if it puffs during baking. Let cool.

Step 4

Stir the egg yolks, crème fraîche, Gruyère, thyme, 1/2 teaspoon of salt and 1/4 teaspoon of pepper into the mushrooms and squash. Spread the mixture on the pastry inside the border. Bake for 15 minutes, until the cheese is melted. Cut into squares and serve.