Put the pumpkin puree in a 2-quart saucepan over medium heat and bring to a simmer. CookThe puree is cooked to evaporate some of the moisture and concentrate the sugars., stirring occasionally, until slightly thickened, 2 to 3 minutes. Remove from heat and add the ginger, nutmeg, and dark brown sugar. Set aside to cool to room temperature, approximately 1 hour.
Heat oven to 300°F.
Cut a parchment paper circle and a long strip to fit the bottom and sides of a 9-by-3-inch cake pan. Brush the bottom and sides of the pan with melted butter. Adhere the parchment pieces to the bottom and sides of the pan.
Prepare the crust by placing the ginger cookies, dark brown sugar, and salt into the bowl of a food processor and process into a fine crumb, 1 to 2 minutes.
With the food processor running, drizzle in the melted butter. Once all the butter has been added, stop the food processor and pulse until the butter is well-incorporated, 3 to 4 pulses.
Press the crust mixture into the bottom of the parchment-lined pan using a weighted glass. Bake for 15 minutes. Then cool for at least 10 minutes on a cooling rack.
Combine the eggs and vanilla in a small mixing bowl. Bring two quarts of water to a boil in an electric kettle.
Beat the cream cheese in a stand mixer with the paddle attachment for 10 seconds on low speed. Add the prepared pumpkin mixture and mix on low for 30 seconds. Stop and scrape down the sides of the bowl. Turn the mixer up to medium, and beat until the mixture is lump free, 1 to 2 minutes.
With the mixer still on medium speed, slowly pour in half of the egg mixture. Stop and scrape down the sides of the bowl. Add the rest of the egg mixture and mix until completely combined. Pour the filling into the cooled crust.
Line a roasting pan with a kitchen towel. Place the cheesecake to one side of the roasting pan and pour the boiling water in on the other side. Center the cheesecake in the roasting pan and bake for 2 hours. Turn the oven off and open the door for 1 minute. Close the door for 30 minutes.
Remove the cheesecake from the water bath and place in the refrigerator for 6 hours to completely cool before serving. Do not attempt to remove the water bath with the cheesecake in it from the oven, unless you like water in your cheesecake.
When ready to serve, place the entire cake pan into a hot water bath for about 15 seconds. Run a small offset spatula around the cheesecake to free the parchment and remove it. Place a piece of wax paper atop the cake and invert a large plate or the bottom of a spring-form pan on top of it. Flip the whole cheesecake over and remove the cake pan. Remove the parchment paper and place a cardboard cake circle or large plate on the bottom of the cake. Flip the whole thing over again and remove the wax paper.
To slice, place your knife (the longest and thinnest blade you've got) into a hot water bath and wipe dry each time you make a pass through the cake. Cut into 12 slices.
Store in the fridge, covered, for up to 1 week.