Heat oven to 325°F and arrange rack in the middle. Grease a 9-inch springform pan.
Place gingersnaps in the bowl of a food processor fitted with the blade attachment and pulse until cookies are fine crumbs (you should have about 2 cups). Stop the food processor and pour in the melted butter, pulse a couple more times until the crumbs are moistened.
Pour the crumb mixture into the springform pan and press evenly over the bottom and up the sides of the pan. Bake until the crust is darker and fragrant, about 15 to 20 minutes. Let crust cool on a wire rack while you make the cheesecake batter.
Combine pumpkin, cinnamon, salt, nutmeg, and cloves in a small bowl and set aside. Combine sugar and flour in another bowl and whisk to break up any lumps.
Beat the cream cheese in the bowl of a stand mixer fitted with the paddle attachment until light and fluffy, about 3 minutes. Pour in the sugar mixture and beat until incorporated. Stop the mixer and scrape down the sides of the bowl and the paddle.
Add the vanilla seeds and 1 egg yolk and beat on low until incorporated. Add remaining yolk and whole eggs, one at a time, letting the first one incorporate fully before adding the next. Stop the mixer and scrape down the sides and paddle again. Add pumpkin mixture and beat until fully incorporated.
Pour cheesecake batter into the pan then gently rap the pan on the counter to free any air bubbles. Bake cheesecake until the edges are lightly browned and the center is barely set, about 55 to 65 minutes. Turn oven off and let cake cool in the oven for 1 hour.
Remove the cheesecake from the oven and place on a cooling rack. Place a baking sheet over the cheesecake. Let it cool, removing the baking sheet a few times to wipe off any condensation that may gather on the surface, until the pan is lukewarm to the touch, about 1 1/2 hours. Remove the baking sheet and refrigerate the cake until chilled, at least 4 hours or overnight. To serve, run a thin knife between the cake and the pan then unlock the sides of the pan, slice and serve.