Make the crust: Pulse the flour, parmesan, sugar and 1/2 teaspoon each fine salt and pepper in a food processor. Pulse in the shortening until the mixture looks like coarse meal. Add the butter and pulse until pea-size pieces form. Add 2 tablespoons beaten egg (reserve the rest) and 2 tablespoons ice water; pulse a few times until the dough begins to clump together, adding more water 1/2 tablespoon at a time as needed. Transfer the dough to a sheet of plastic wrap, press into a rectangle, wrap tightly and refrigerate until firm, at least 1 hour or overnight.
Meanwhile, make the filling: Preheat the oven to 450˚ F. Toss the mushrooms with 1 tablespoon olive oil and a pinch each of kosher salt and pepper on a rimmed baking sheet. Roast until the mushrooms shrink and release some liquid, 8 to 10 minutes. Add the celery root, parsnips and/or turnips, the carrots, fennel, shallots, rosemary, 1 teaspoon kosher salt, 1/2 teaspoon pepper and the remaining 2 tablespoons olive oil; toss. Roast until the vegetables are tender and the mushroom liquid has evaporated, about 30 minutes.
Lightly dust a piece of parchment paper with flour. Roll out the dough on the parchment into a 12-by-14-inch rectangle. Refrigerate until the vegetables are done.
Drizzle 2/3 cup broth on the baking sheet, especially in the corners, and scrape up any browned bits. Stir in the remaining 1 cup broth, the heavy cream and peas.
Cut the dough into 3/4-inch-wide strips. Lay half the strips across the top of the sheet pan diagonally, trimming or adding extra bits as needed. Lay the remaining strips on top in the other direction. Brush the dough with the reserved beaten egg. Bake until the crust is golden and the liquid on the pan thickens, 25 to 30 minutes. Let cool 5 minutes before serving.