Preheat the oven to 375 degrees F.
Cut the lid off the pumpkin and scoop out the seeds and fibers. Scrape away some of the flesh, especially around the top, to form a large, smooth inner cavity. Make sure to leave the shell at least 1-inch thick all around. Oil the pumpkin inside and out and season inside with salt and pepper. Replace the lid. Place on a baking sheet and bake until about 2/3 cooked but still firm enough to stand and act as the soup container, about 40 minutes. Do not over-bake. If a good deal of liquid gathers in the bottom, spoon out and discard. Set aside; it will stay warm on its own for a good while.
Bring the stock to a low simmer in a saucepan. Heat the remaining 2 tablespoons olive oil in a large saute pan over medium-high heat until hot. Add the onion and cook until soft but not brown, 2 to 3 minutes. Add the thyme, stir, and add the pastina. Stir and add 2 cups of heated stock. Bring the mixture up to a simmer.
Simmer, adding the stock 1/2 cup at a time as the previous addition is absorbed and stirring occasionally to prevent the pastina from sticking to the bottom of the pan, until the pasta is al dente, about 15 minutes. Add the roasted squash and turkey to reheat. Stir well. The consistency should be quite loose, like a thick soup. Add more stock, if necessary. Add the 1/2 cup cheese and let melt for a moment before stirring in. Taste for seasoning.
Fill the reserved pumpkin shell with the pastina. Grate Parmesan on top and replace the lid and take to the table immediately.
You could also serve this in individual tureens using smaller pumpkins or acorn squash.
Preheat the oven to 400 degrees F.
Peel the squash with a vegetable peeler. Halve lengthwise, discard the seeds, then cut into 1-inch dice. Place in a large bowl and season with salt and pepper. Heat the butter in a medium skillet over medium-high heat. When the butter ceases to foam and has turned a light brown immediately add the sage, vinegar, sugar (stand back so as not to get splattered), and molasses. Mix well and let simmer over medium-low heat for 1 to 2 minutes to meld the flavors.
Pour the vinegar mixture over the squash and toss well. Season with salt and pepper then transfer to a heavy rimmed baking sheet or baking dish large enough to hold the squash in a single layer. Place in the preheated oven and roast, tossing at least once, until very tender and caramelized, about 20 minutes. Set aside until cool enough to handle but still warm, so the liquids are runny.
Working in batches, if necessary, transfer the warm squash and all the cooking liquids to a food processor and process until smooth. Use immediately, refrigerate for up to 5 days, or freeze for up to 2 months.