Preheat your oven to 350 degrees F.
In a microwave-safe bowl, melt ½ cup of butter in 30-second increments.
In a mixing bowl, combine the graham cracker crumbs, melted butter, and ¼ cup of granulated sugar.
Press this graham cracker mixture into the bottom of an 8x8 inch square pan and bake at 350 degrees for 10 minutes.
Once removed from the oven, allow the crust to cool.
In a large mixing bowl, beat the cream cheese until smooth.
Add the eggs, lemon zest, lemon juice, and vanilla, and mix until well combined.
Pour this mixture onto the prepared crust and spread it out evenly.
Bake at 350 degrees for 25-30 minutes, or until the cheesecake is set and slightly jiggly.
Allow the cheesecake to cool completely, then refrigerate for 6-8 hours or overnight.
Once chilled, cut into squares and serve. For an optional garnish, top with whipped cream and a fresh lemon slice.