Preheat your oven to 350°F. In a medium bowl, combine the graham cracker crumbs, 1/4 cup of granulated sugar, and melted butter until the mixture resembles wet sand and holds together when pinched.
Firmly press the crumb mixture into the bottom and about 1 inch up the sides of a 9-inch springform pan. Bake the crust for 8 minutes, then set it on a wire rack to cool.
In the bowl of a stand mixer fitted with the paddle attachment, beat the room-temperature cream cheese and 1 1/4 cups of granulated sugar on medium speed until the mixture is completely smooth and creamy, about 3 minutes.
Reduce the mixer speed to low and add the eggs one at a time, mixing just until each yolk disappears into the batter before adding the next. Scrape down the sides of the bowl with a rubber spatula.
Add the sour cream, key lime juice, key lime zest, and vanilla extract. Mix on low speed until all ingredients are just combined and the filling is uniform, being careful not to overmix.
Pour the filling into the cooled crust. Place the springform pan on a baking sheet and bake for 50 to 60 minutes, or until the edges are set but the center still has a slight jiggle when the pan is gently shaken.
Turn off the oven, crack the door open, and let the cheesecake cool inside for 1 hour. This gradual cooling helps prevent cracks from forming.
Remove the cheesecake from the oven and run a thin knife around the inner edge of the pan to loosen it. Let it cool completely at room temperature, then refrigerate for at least 4 hours or, ideally, overnight before serving.