Step 1

Heat oven to 425 degrees. Bring 2 cups of water to boil in a small pot. Place the porcini mushrooms in a medium bowl and pour the boiling water on top; set aside to soak and soften.

Step 2

Meanwhile, in a large (12-inch) ovenproof skillet, heat 2 tablespoons oil over medium-high. Working in two batches so the mushrooms brown evenly, add half the oyster mushrooms, season with salt and pepper and cook, undisturbed, until browned underneath, about 3 minutes. Stir the mushrooms, then continue to cook until browned all over, about 3 minutes. Transfer to a medium bowl. Repeat with 2 tablespoons oil and the remaining oyster mushrooms; transfer them to the medium bowl. Add another 2 tablespoons oil, then add the cremini mushrooms, season with salt and pepper and cook, stirring occasionally, until browned and tender, 6 to 8 minutes, lowering the heat as needed to avoid burning. Transfer to the bowl.

Step 3

Reduce the heat to medium and add the butter to the skillet. Once the butter is melted, add the onion and cook until starting to soften, about 5 minutes. Add the kale, garlic, thyme and rosemary, season with salt and pepper and cook, stirring occasionally, until softened, about 3 minutes, adding 1 to 2 tablespoons oil if the pan seems dry. Sprinkle with the flour and cook, stirring, until flour is dispersed and lightly toasted, 2 minutes.

Step 4

Stirring constantly, gradually add the stock, cream and 3/4 cup of the liquid from the soaking porcini mushrooms. Cook over medium heat, stirring occasionally, until thickened, about 5 minutes. Remove from heat, then stir in the carrots, parsnips, potatoes, and the sautéed mushrooms and any accumulated juices. Strain the soaked porcini mushrooms, discarding any remaining liquid, then roughly chop the mushrooms and stir them into the filling. Season to taste with salt and pepper. (If making ahead, the filling can be refrigerated a day in advance; just warm it up slightly before assembling and baking.)

Step 5

On a lightly dusted work surface using a lightly floured rolling pin, roll out the puff pastry into a roughly 14-inch circle. Transfer to the top of the skillet, allowing slight overhang, folding under and pinching any overhang just below the rim of the skillet. In a small bowl, whisk the egg with 1 teaspoon water. Lightly brush the egg wash over the top of the puff pastry, then cut 5 (2-inch) slits into the puff pastry, starting near the center and radiating outward, like the rays of the sun, or create a stripe or crisscross pattern in the puff pastry by gently scoring it with a paring knife without fully cutting through. Crimp the edges using the tines of a fork, if desired.

Step 6

Set the skillet on top of an aluminum foil-lined baking sheet and transfer to the middle rack of the oven. Bake until the puff pastry is puffed and golden, about 30 minutes. Let sit 10 minutes before serving.