Preheat your oven to 350 degrees F. Wrap the outside of a 9-inch springform pan tightly with heavy-duty aluminum foil, covering the bottom and sides completely.
Prepare the crust: Mix graham cracker crumbs, 1/4 cup sugar, and melted butter until evenly moistened. Press the mixture firmly and evenly into the bottom of the prepared springform pan.
Bake the crust for 10 minutes. Remove from the oven and let it cool while you prepare the filling. Reduce the oven temperature to 325 degrees F.
Prepare the filling: In a large bowl, beat the softened cream cheese with an electric mixer on medium speed until completely smooth. Scrape down the sides of the bowl.
Add 1 1/2 cups sugar and flour to the cream cheese mixture. Beat until just combined. Do not overmix.
Mix in vanilla extract and salt. Beat in the heavy cream until smooth.
Add the eggs and the extra egg yolk one at a time, mixing on low speed after each addition only until the yolk disappears into the batter. Stop mixing as soon as the last yolk is incorporated.
Pour the filling over the cooled crust. Smooth the top with a spatula.
Bake using a water bath: Place the foil-wrapped springform pan inside a larger roasting pan. Carefully pour hot water into the roasting pan until the water reaches about halfway up the sides of the springform pan.
Bake for 60 to 75 minutes. The edges should look set, but the center 2 inches should still wobble slightly when gently nudged.
Turn off the oven. Prop the oven door open slightly (about 2 inches) using a wooden spoon. Let the cheesecake cool inside the turned-off oven for 1 hour. This slow cooling prevents cracking.
Remove the cheesecake from the water bath and remove the foil. Let it cool completely on a wire rack at room temperature.
Once cool, cover the pan loosely with plastic wrap and refrigerate for at least 8 hours, or preferably overnight, before serving.
Notes
Use room temperature ingredients for the filling to prevent lumps and ensure a smooth texture.
Do not overbeat the batter once the eggs are added; this incorporates air which causes cracking during cooling.
The water bath provides moist, even heat, which is key to a creamy, dense texture.