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FOR SQUASH:
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1. Preheat the oven to 375 and set two oven racks evenly apart. Cut each squash in -half lengthwise and scoop out and discard the seeds and stringy pulp. Place 1 1/2 teaspoons of butter inside each squash half and season with salt and pepper. Place the squash halves cut-side down in a baking dish large enough to hold all four (divide them among two baking dishes if they do not fit in one). Add enough water to go 1/2 inch up each squash half. Bake the squash until the skins start to brown and the flesh is tender, about 45 minutes. Remove from the oven (leaving the oven on), flip over so the flesh is exposed, and set aside to cool. When the squash halves are cool enough to handle, scoop out the flesh and discard the skins, reserving the flesh and baking liquid.
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2. Toss the pumpkin seeds with the vegetable oil, 1/4 teaspoon salt, and the thyme, then spread them out on a baking sheet. Bake until just lightly browned, 8 to 10 minutes. Remove and let cool.
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3. Set a large soup pot over medium-low heat and add the olive oil. Add the onion and garlic and sweat by cooking them until the onions are translucent, about 5 minutes. Add the apple and fennel, season with salt and pepper, and sweat until the fennel is translucent, about 15 minutes more.
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4. Add the wine and simmer to reduce almost all of the liquid. Add the broth, cider, reserved squash liquid, and the squash flesh. Bring them to a boil, then reduce the heat and simmer until the squash is very tender, about 1 1/2 hours.
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5. Meanwhile, slowly melt the remaining 1/4 cup butter in a heavy-bottomed pot over medium heat. Continue heating until the milk solids begin to fall to the bottom of the pot and start to brown. Take care not to let the solids blacken, removing the butter from the heat when it is dark after 10 to 15 minutes. Strain the brown butter through a fine-mesh sleeve and discard the solids.
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6. Working with a blender, puree the soup until smooth. Add the brown butter slowly, stopping to loosen the ingredients with a wooden spoon if necessary. Add additional broth to thin the soup if it's too thick. Adjust the seasoning with salt and pepper.
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7. Ladle the soup into bowls, drizzle with sherry gastrique (recipe below), and sprinkle with the toasted pumpkin seeds.
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FOR SHERRY GASTRIQUE:
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1. Combine the vinegar, sugar, thyme, and peppercorns in a small heavy-bottomed nonreactive pot. Bring to a boil, then reduce the heat to medium-low. Summer the liquid to reduce until syrupy, 8 to 10 minutes.
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2. Cool the liquid and strain out the thyme and peppercorns. You can store in a tightly sealed container (no need to refrigerate) for up to 1 month.
