1. To make the filling: Prepare a steamer by filling a large round casserole with water to a depth of 3 inches; the casserole should be able to hold the squash comfortably and have a tightly fitting lid. Put a steamer rack or enough crumpled heavy-duty aluminum foil to support the squash on the bottom; the rack or foil should be just above the waterline. Set over medium heat and bring to a steady simmer. Put the whole squash on the rack, cover the pot, and steam until a knife pierces the flesh easily, about 1 hour.
2. Remove from the heat, uncover the pot, and cool the squash in the steamer until cool enough to handle.
3. Meanwhile, make the crust: Preheat the oven to 325°F. Lightly butter a 9-inch spring form pan, line with parchment paper, and butter the paper. Set aside.
4. Spread the walnuts on a rimmed baking sheet and toast, shaking the pan occasionally, until fragrant, about 15 minutes. Cool completely. Turn the oven down to 300°F.
5. Put the walnuts and ¼ cup of the brown sugar in the bowl of a food processor and pulse until the walnuts are coarsely ground. (You can also crush the walnuts by putting them in a heavy-duty plastic bag and gently pounding with a heavy skillet.) Transfer the walnuts to a mixing bowl and add the graham cracker crumbs, lime zest, cinnamon, ginger, salt, and the remaining ¼ cup brown sugar. Mix well, then add the melted butter and mix with your hands until everything is evenly moistened. Transfer the mixture to the prepared pan and press into an even layer on the bottom.
6. Bake the crust until golden brown, about 12 minutes. Cool completely. Leave the oven on.
7. When the squash is cool enough to handle, remove it from the steamer, cut it in half, and scoop out and discard the seeds and strings. Scoop out 2½ cups of the squash flesh into a small bowl. Reserve any remaining squash for another use.
8. Put the cream cheese, sugar, cinnamon, ginger, nutmeg, and salt into the bowl of a food processor and process, scraping down the sides and bottom of the bowl occasionally, until the mixture is light and smooth. Add the squash and process again, scraping down the sides and bottom of the bowl occasionally, until smooth. Add the brandy and eggs and process just until they are incorporated. Transfer to a bowl and finish mixing with a rubber spatula.
9. Transfer the filling to the cooled crust. Bake until the center is set but still slightly jiggly, about 1 hour. Cool completely and unrnold.
10. Serve the cheesecake with the condensed milk chantilly, if desired.