For the tofurkey: Place a colander inside a large bowl and line it with two layers of ultrafine cheesecloth; set aside. Using your hands, break the tofu into roughly 2-inch pieces. Place about half of the tofu in a food processor fitted with a blade attachment. Add the miso, salt, thyme, pepper, soy sauce, and sage and process, scraping down the sides of the bowl as needed, until the mixture is very smooth, about 2 minutes total. Transfer to a large bowl. Process the remaining tofu until smooth. Transfer to the bowl with the tofu-miso mixture and stir until evenly combined. Transfer to the colander, smooth the top, and fold the cheesecloth over to completely cover the tofu mixture. Set a plate about the size of the inside of the colander on top of the cheesecloth and place a few heavy cans or jars on top of the plate to weigh down the tofu (the weight helps expel any excess water). Refrigerate overnight, or at least 12 hours. For the glaze: In a small bowl, whisk together all ingredients until combined. Cover and store in the refrigerator until ready to use. For the stuffing: Place oil in a medium saucepan with a tightfitting lid and set over medium heat. When it shimmers, add shallots, celery, thyme, and sage, season with salt and pepper, and cook, stirring occasionally, until shallots just begin to soften, about 3 minutes. Add mushrooms and cook, stirring occasionally, until softened, about 3 minutes more. Add brown rice and stir to evenly combine. Add broth and water and bring to a boil. Reduce heat to low, cover, and simmer until rice is tender, about 50 minutes. (If the rice has not absorbed all of the liquid, continue to cook, uncovered, until the liquid is absorbed, then replace the lid.) Remove from heat and let sit, covered, for about 10 minutes. Fluff the rice with a fork. Stir in parsley and vinegar. Taste and season with salt and pepper as needed; set aside to cool slightly. To assemble and serve: Heat the oven to 375°F and arrange a rack in the middle. Coat a 9-inch springform pan with oil and place on a baking sheet. Remove the colander from the refrigerator. Remove the weights and plate and peel away the cheesecloth from the top of the tofu. Using a 1-cup measure, transfer 4 cups of the tofu mixture from the colander to the springform pan and, using a rubber spatula, spread into an even layer. Arrange the stuffing in an even layer over the tofu, leaving a 1/2-inch border. Spread the remaining tofu in an even layer over the stuffing, completely covering it, and smooth it to the edge of the pan. Using a pastry brush, coat the top with a thin layer of the glaze. Bake in the oven until the top is dark golden brown, about 75 minutes, brushing the top with glaze every 15 minutes. Remove from the oven and let cool on the baking sheet, about 10 minutes. Run a knife around the inside of the pan before removing the outer ring. Cut into wedges and serve, if you choose, with mushroom gravy.
