Preheat your oven to 325°F (160°C). Wrap the outside of a 9-inch springform pan tightly with heavy-duty aluminum foil.
Prepare the crust: Combine graham cracker crumbs, 1/3 cup sugar, and melted butter. Press the mixture firmly into the bottom of the prepared pan. Bake for 10 minutes. Let cool slightly.
Prepare the cheesecake filling: Beat the softened cream cheese until smooth. Gradually beat in 3/4 cup sugar until combined. Beat in the eggs one at a time, mixing just until incorporated after each addition. Mix in the vanilla extract.
Pour the cheesecake batter over the cooled crust. Place the springform pan into a larger roasting pan. Carefully pour hot water into the roasting pan until it reaches halfway up the sides of the springform pan (this creates a water bath).
Bake for 55 to 65 minutes, or until the edges are set but the center still jiggles slightly.
Turn off the oven, crack the door open, and let the cheesecake cool inside the oven for 1 hour. Remove from the water bath and cool completely on a wire rack. Chill in the refrigerator for at least 6 hours, or preferably overnight.
Prepare the topping: When ready to serve, combine the heavy cream and 1/2 cup sugar in a small saucepan. Heat over medium heat, stirring constantly, until the sugar dissolves and the mixture begins to simmer and lightly brown (about 5-7 minutes). This creates a custard layer.
Pour the warm custard mixture evenly over the chilled cheesecake. Return the cheesecake to the refrigerator to chill for at least 2 hours until the custard layer is firm.