Put the graham crackers in a food processor and blend until they are fine crumbs.
Mix the graham cracker crumbs, melted butter, and ¾ cup sugar in a medium bowl.
Line a 9x9-inch baking pan with parchment paper and spread the graham cracker mixture into the bottom of the pan, pressing firmly. Bake this for 5 minutes.
Add the cream cheese to a large bowl and mix with a hand mixer until smooth. Add the rest of the sugar, lemon juice, lemon zest, and vanilla. Beat until well mixed. Then fold in the beaten egg.
Reduce the oven's heat to 300 F.
Pour the filling mixture over the crust, then bake for 35 minutes. The top will look lightly browned, but the center will remain jiggly. Cool in the pan for 1 hour, then refrigerate for 8 hours.
Lift the parchment out of the pan and cut into 9 large squares or 16 mini squares. Top with fresh lemon zest, powdered sugar, and lemon slices.