Heat the oven to 425°F (220°C). Line a 9-inch springform pan with two crumpled sheets of parchment paper, pleating them up the sides so the paper rises about 2 inches above the rim.
Beat the cream cheese and sugar in a large bowl on medium speed until completely smooth and creamy, about 2 minutes, scraping down the bowl halfway through.
Beat in the pumpkin puree, vanilla, pumpkin pie spice, cinnamon, and salt until smooth.
Add the eggs one at a time, beating just until each disappears. Do not overbeat — extra air causes cracking and a puffy, sunken center.
Sift in the flour and beat on low just until no dry streaks remain.
Pour in the heavy cream and mix on low until the batter is silky and pourable.
Pour the batter into the prepared pan and smooth the top. Tap the pan firmly on the counter twice to pop any large air bubbles.
Bake at 425°F (220°C) for 45–55 minutes, until the top is deeply browned — dark mahogany in spots — and the center still jiggles like set gelatin when you shake the pan gently. A burnt-looking top is correct; that is the Basque signature.
Cool in the pan on a rack for 1 hour. The cake will sink and wrinkle as it cools — that is normal and expected.
Chill uncovered (or loosely covered) in the refrigerator for at least 4 hours, preferably overnight, before slicing with a hot, dry knife.