Preheat the oven to 325ºF. In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese and both sugars on low speed until smooth and creamy. Scrape down the sides of the bowl, add the vanilla, cornstarch, salt, and spices, and continue beating on low until smooth. Scrape down the sides of the bowl, and mix in the sour cream on low.
Pour 4 cups of water into a medium saucepan, and place over high heat. Allow it to come to a boil as you finish the rest of the batter. Place a large roasting pan on the bottom rack of the oven.
To the batter, add the eggs and egg yolks, two at a time, beating on low until combined--try not to over-mix. Add the pumpkin puree, and mix on low until just combined. Remove bowl from mixer and mix the rest by hand, scraping the sides and bottom of the bowl for even incorporation.
Pour batter into the crust, smoothing out the top. Open the oven, and carefully pour the boiling water into the roast pan--it might sizzle, so be extra cautious. Place the cheesecake on the rack directly above the roasting pan, and close the oven to trap the steam inside.
Bake cheesecake for 70-75 minutes, or until puffed and center jiggles like jell-o when lightly shaken. If you're unsure, you can use an instant-read thermometer--it should be at least 150ºF, but not over 165ºF. Turn off the oven, crack open the oven door, and allow cheesecake to sit in the oven for 1 hour.
After an hour, let the cheesecake cool at room temperature for another hour before placing in the refrigerator to chill uncovered until it's room temperature or cooler. Cover with a piece of foil or plastic wrap and chill for at least 8 hours, preferably overnight.