Make the crust: Preheat oven to 350°F (175°C). In a medium bowl, combine the graham cracker crumbs, granulated sugar, and melted butter. Mix well until the crumbs are evenly moistened. Press the mixture firmly into the bottom of a 9-inch springform pan. Bake for 8-10 minutes, until lightly golden. Let cool completely.
Make the filling: In a large bowl, beat the softened cream cheese until smooth and creamy. Gradually add the granulated sugar, beating on low speed until combined. Then, increase the speed to medium and beat until light and fluffy.
Add peanut butter: Beat in the creamy peanut butter until well combined.
Add eggs and vanilla: Beat in the eggs one at a time, mixing on low speed after each addition. Stir in the vanilla extract.
Bake the cheesecake: Pour the peanut butter filling into the prepared graham cracker crust. Bake for 50-60 minutes, or until the edges are set and the center is just slightly jiggly.
Cool and chill: Turn off the oven and leave the cheesecake in the oven with the door slightly ajar for 1 hour to cool slowly. Then, remove the cheesecake from the oven and let it cool completely to room temperature. Cover the pan with plastic wrap and refrigerate for at least 4 hours, or preferably overnight, to allow the cheesecake to set completely.
Garnish and serve: Before serving, remove the sides of the springform pan. Top the cheesecake with your favorite toppings, such as chocolate ganache, chopped peanut butter cups, whipped cream, chopped peanuts, or caramel sauce.