Step 1: Preheat oven to 350°F (175°C). Lightly grease a 9-inch springform pan.
Step 2: In a medium bowl, combine graham cracker crumbs, 1/4 cup granulated sugar, and melted butter. Mix until well combined.
Step 3: Press the crumb mixture evenly into the bottom of the prepared springform pan. Bake for 8-10 minutes, or until lightly golden. Remove and let cool completely on a wire rack.
Step 4: Reduce oven temperature to 325°F (160°C). Prepare a water bath: Wrap the bottom and sides of your springform pan tightly with several layers of heavy-duty aluminum foil.
Step 5: In a large mixing bowl, beat softened cream cheese with an electric mixer on medium speed until smooth and creamy, about 2-3 minutes. Ensure there are no lumps.
Step 6: Gradually add 1 3/4 cups granulated sugar and the all-purpose flour to the cream cheese, beating on low speed until just combined and smooth. Scrape down the sides of the bowl as needed.
Step 7: Beat in the sour cream and vanilla extract until just incorporated.
Step 8: Add the eggs one at a time, beating on low speed until *just* combined after each addition. Do not overbeat once eggs are added.
Step 9: Pour the cheesecake filling over the cooled graham cracker crust in the prepared springform pan.
Step 10: Place the foil-wrapped springform pan into a larger roasting pan. Carefully pour hot water into the roasting pan until it reaches about halfway up the sides of the springform pan.
Step 11: Carefully transfer the roasting pan to the preheated oven. Bake for 60-75 minutes, or until the edges are set, but the center (about a 2-3 inch circle) still jiggles slightly when gently shaken.
Step 12: Turn off the oven, crack the oven door open slightly, and leave the cheesecake in the water bath for 1 hour.
Step 13: Carefully remove the cheesecake from the water bath and roasting pan. Remove the foil. Let it cool completely on a wire rack at room temperature for another 1-2 hours.
Step 14: Once completely cool, cover the cheesecake loosely with plastic wrap and refrigerate for at least 6 hours, or preferably overnight, before serving.