Adjust oven racks to upper-middle and lower-middle positions and preheat oven to 425°F (220℃). If using shiitake mushrooms, remove and discard stems; for portobello and cremini mushrooms remove and reserve stems, discarding any woody stems. Cut shiitake and portobello mushrooms into ½-inch pieces. Halve cremini mushrooms or quarter if larger than 1 ½ inches wide.

In a medium bowl, toss mushrooms with oil until coated. Evenly divide between 2 large rimmed baking sheets and spread into an even layer. Roast mushrooms, rotating pans top to bottom halfway through, until lightly browned, about 20 minutes.

Meanwhile, in a medium saucepan over medium-high heat, bring 2 ½ cups water, porcini mushrooms, and reserved cremini and portobello mushroom stems to a boil; reduce heat to medium, simmer mushrooms, stirring occasionally, until mushrooms are softened, 3 to 5 minutes. Pour mushroom mixture into a blender. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening, then process until smooth, 1 to 2 minutes; set aside.

In a 10-inch cast iron skillet, melt butter over medium heat, add onions and cook, stirring occasionally, until translucent, 3 to 4 minutes. Add carrots and parsnips and cook, stirring occasionally, until carrots and parsnips are beginning to soften, about 5 minutes. Stir in thyme, miso, oregano, salt, and pepper and cook, stirring occasionally, until fragrant, about 1 minute. Sprinkle flour over vegetable mixture and cook, stirring constantly, until flour is lightly toasted, about 2 minutes.

Gradually stir in blended mushroom mixture and milk until combined. Bring to a simmer over medium heat, then continue to simmer, stirring occasionally, until slightly thickened, about 2 minutes. Remove from heat and stir in roasted mushrooms; set aside.

Working on a lightly floured work surface, roll puff pastry sheet into an 11-inch square. Carefully place on top of hot mushroom filling; fold excess dough underneath itself and crimp as desired just below rim of skillet so it's flush with the skillet's edge. Using a pastry brush, lightly brush dough with egg wash.

Using a paring knife, lightly score the entire surface of the pastry in a diagonal crosshatch pattern. Make sure not to cut all the way through the dough. Cut five ¼-inch wide steam vents into the dough, spread out evenly over the pie.

Place skillet on a large rimmed baking sheet. Bake at 425°F (220℃) on the lower-middle rack until pastry is puffed and golden and sauce is bubbly, about 25 minutes. Let cool for 5 minutes before serving.