Preheat oven to 350 degrees. Make the filling: Cut squash in half lengthwise; remove seeds, and reserve for another use. Place squash, cut sides down, on a baking sheet lined with buttered parchment paper. Bake until tender, about 45 minutes. Transfer sheet to a wire rack; let cool.
Make the crust: Put butter and sugar into the bowl of an electric mixer fitted with the paddle attachment; mix on medium speed until pale and fluffy. Mix in egg yolk and vanilla. Reduce speed to low. Add flour and salt; mix until dough comes together. Shape into a disk; wrap in plastic. Refrigerate until firm, at least 30 minutes and up to 1 day.
On a lightly floured surface, roll dough into a 10-inch circle, a scant 1/4 inch thick. Fit into bottom of a 10-inch springform pan. Freeze 15 minutes. Bake until crust is firm and pale golden, 12 to 15 minutes. Transfer to a wire rack; let cool completely. Reduce oven temperature to 325 degrees.
Scoop squash flesh into the bowl of a food processor; process until pureed. Transfer 1 cup puree to a medium bowl; reserve remainder for another use. Stir in cinnamon, allspice, nutmeg, and ginger.
Wrap exterior of springform pan (including base) in 2 layers of foil. Butter sides of pan; set aside. Put cream cheese into the clean bowl of an electric mixer fitted with the clean paddle attachment; mix on medium speed until fluffy. Gradually add sugar and flour, mixing until smooth. Mix in sour cream, vanilla, and salt. Mix in eggs, 1 at a time, until just combined; do not overmix.
Stir 2 cups cream cheese mixture into squash mixture; set aside. Pour remaining cream cheese mixture into prepared pan on top of the crust. Drop dollops of squash-cream cheese mixture on top. Gently swirl with a butter knife.
Set pan in a large, shallow roasting pan. Transfer to oven. Carefully pour enough boiling water into roasting pan to come halfway up sides of cheesecake. Bake until cake is set but still slightly wobbly in center, 50 to 60 minutes. Turn off oven; let stand in oven with door slightly ajar 1 hour. Let cool completely on a wire rack. Refrigerate at least 6 hours or overnight. Run a knife around sides of cake; unmold.