Preheat the oven to 350 degrees F.

Place the squash halves, cut sides up, in a large baking dish. Place 1 teaspoon of the butter in the center of each squash. Sprinkle the insides lightly with the cinnamon and nutmeg. Add enough water to cover the bottom of the pan by about 1/4-inch (about 1/2 cup), cover tightly with aluminum foil, and roast until the squash is tender, 50 minutes to 1 hour.

Remove the squash from the oven, uncover, and let sit until cool enough to handle. Using a spoon, scoop the flesh from the squash into the bowl of a food processor..

Melt the remaining 3 tablespoons butter in a medium skillet over medium-high heat. When the butter is foamy, add the shallots and cook, stirring, until soft and fragrant, 45 seconds to 1 minute. Add the apples, salt, and pepper and cook, stirring occasionally, for 3 minutes. Add the stock and cook, stirring occasionally, until the apples are very tender, 8 to 10 minutes.

Remove the pan from the heat and transfer the apple mixture to a food processor with the squash. Puree the mixture on high speed until smooth. Adjust the seasoning, to taste, and serve warm.