Position the oven racks in the center of the oven and preheat it to 350 degrees Fahrenheit.
In a medium bowl, mix together graham cracker crumbs, 5 tablespoons of sugar, and melted butter until the mixture resembles damp sand. Press the crust mixture firmly into the bottom and halfway up the sides of a 9-inch springform pan using the bottom of a measuring cup. Bake for 7 minutes, then remove from the oven and set aside.
Lower the oven temperature to 325 degrees Fahrenheit. In a large bowl or the bowl of a stand mixer, beat the cream cheese for 30 seconds until smooth. Scrape the sides and bottom of the bowl, then add in 1 1/4 cups of granulated sugar and mix until smooth.
Incorporate the vanilla extract, pure pumpkin, pumpkin pie spice, and cinnamon into the mixture. Scrape the sides and bottom of the bowl again and mix to ensure even blending of the cheesecake batter.
Crack the eggs into a liquid measuring cup and whisk them until well scrambled. With the mixer set to low speed, gradually pour the beaten eggs into the cream cheese mixture, stopping as soon as the eggs are incorporated. Remove the bowl from the mixer and scrape the sides and bottom again to ensure a smooth mixture. If there are small lumps, gently fold them in with a rubber scraper.
Once the batter is smooth, tap the bowl on the counter for 30-45 seconds to release as many air bubbles as possible; you should see bubbles popping on the surface. Carefully pour the filling into the center of the graham cracker crust and gently smooth the top, noting that it will be very full.
Bake at 325 degrees Fahrenheit for 40 minutes. Then reduce the temperature to 250 degrees Fahrenheit and continue baking for an additional 45 minutes. After this time, turn off the oven and keep the cheesecake inside for another 30 minutes to allow for carryover cooking without opening the oven door. Crack the oven door to let the cheesecake cool slowly for one hour before removing it. The cheesecake should be slightly warm at this stage. Allow it to come to room temperature on the counter for 3-4 hours before covering it with plastic wrap and transferring it to the fridge.
Chill the cheesecake in the refrigerator until completely cooled, which may take 6 hours to overnight. When ready to serve, release the springform pan and remove the collar. Decorate as desired. To slice, dip a sharp knife into hot water, wipe off any excess, and cut the cheesecake. For clean slices, it's recommended to dip the knife in water between each cut.
