Heat the oven to 400°F.
In a large bowl, whisk together 2 cups plus 2 Tbs. of the flour, all of the sage, and 1/2 tsp. salt. Add the 3 Tbs. of chilled butter and all of the shortening. Use your fingers to crumble and massage the ingredients together until they are well mixed and the consistency of coarse meal. Pour in 6 Tbs. of ice water (just the water, not the ice) and very gently knead the mixture together. As soon as the dough comes together, stop kneading and adding water (don’t overknead the dough or it will become tough and lose its flakiness). You might need another 3 Tbs. of ice water, for a total of 1/2 cup plus 1 Tbs. ice water.
Form the dough into a ball and place it on a large piece of plastic wrap; cover completely with the plastic wrap and flatten the ball into a circle. Refrigerate for about an hour.
Meanwhile, transfer the butternut squash and parsnip slices to a baking sheet with sides and toss with 2 Tbs. oil, 1/2 tsp. salt, and 8 grinds pepper. Mix well with your hands and then roast until tender and golden brown, about 20 minutes. Let cool to room temperature (to expedite this, carefully transfer the roasted vegetables to a plate). Reduce the oven temperature to 375°F.
While the squash and parsnips roast, fill a medium-sized, heavy saucepot two-thirds full of heavily salted water, cover, and bring to a boil over high heat. Once the water is boiling, add the Brussels sprouts, stir, and cook until the vegetables are bright green and tender, about 5 minutes. Use a spider or slotted spoon to drain the vegetables well and transfer them to a bowl. Add 1 Tbs. oil, 1/8 tsp. salt, and 4 grinds pepper, and toss well. Let cool to room temperature.
Add the remaining 2 Tbs. butter to a 10-inch, heavy sauté pan, and heat over medium. Once the butter has melted, add the leeks and 1/4 tsp. salt; stir and sauté until the leeks are very tender, sweet, and just beginning to become very light golden brown in spots, about 8 minutes. Let cool to room temperature (to expedite this, you can place the leeks on a plate and refrigerate).
Sprinkle the remaining 1 Tbs. flour onto a cold, large, clean surface. Unwrap the chilled dough and transfer to the surface. With a very lightly floured rolling pin, roll out the dough into a 1/8-inch-thick circle about 12 or 13 inches in diameter. Transfer to a parchment-lined baking sheet. Evenly sprinkle the center of the dough with Parmigiano-Reggiano (leave a border of 2 to 3 inches all around). On top, evenly sprinkle the leeks (leaving the same border). On top of the leeks, arrange the Brussels sprouts (leaving the same border). Then place the squash and parsnips on top, evenly distributing them, and sprinkle with the Gruyère, leaving the same border.
Fold the edges of the dough over the outer couple of inches of the filling all around in a circle, making a rustic pastry border. Patch holes or weak spots with any excess dough. In a small bowl, whisk together the egg with 1 Tbs. of water. With a pastry brush, brush any exposed dough. Chill the tart in the fridge for 20 minutes.
Bake the tart until the pastry turns light golden brown, about 45 minutes. Drizzle the filling evenly with the truffle oil, if desired. Let the tart cool for 10 to 15 minutes, then slice and serve.
Photo: Kate Sears