Grease an 8-inch springform pan with butter or cooking spray. In a bowl, combine the graham cracker crumbs, melted butter, and 1/4 cup granulated sugar, mixing until the texture resembles wet sand. Press this mixture firmly into the bottom of the prepared pan and up the sides about 2 inches, using the bottom of a measuring cup to create an even layer. Place the crust in the freezer while you prepare the ice cream base—this prevents it from becoming soggy when the ice cream is added later.

Heat the heavy cream, whole milk, and salt in a pot over medium heat until it reaches a gentle simmer. In a separate bowl, whisk together the egg yolks and 1.5 cups granulated sugar until pale and thick, about 2-3 minutes. Slowly add two large spoonfuls of the hot cream mixture to the yolks while whisking constantly to temper them, preventing scrambled eggs. Pour the egg mixture back into the pot with the remaining cream and cook over low heat, stirring constantly, until it coats the back of a spoon (about 170°F if using a thermometer). Strain the hot custard through a fine-mesh sieve into a bowl to remove any cooked egg bits, then let it cool to room temperature for about 15-20 minutes, stirring occasionally.

While the custard cools, bring the room-temperature cream cheese to room temperature if it isn't already, ensuring it's soft enough to blend smoothly. Once the custard has cooled, whisk in the cream cheese until completely smooth and well incorporated—I use a whisk rather than a blender here to avoid overworking the mixture and incorporating too much air. Stir in the lemon juice, lemon zest, vodka (which prevents ice crystals from forming), and vanilla extract. Refrigerate this mixture for at least 6 hours, or ideally overnight, until thoroughly chilled—this is crucial for proper ice cream texture.

While the ice cream base chills, combine the 2 cups sliced strawberries, 0.75 cup sugar, 1 tablespoon lemon juice, and cornstarch in a pot. Let this mixture sit at room temperature for 20 minutes to allow the strawberries to release their juices. Bring the mixture to a boil over medium-high heat, stirring occasionally, and continue cooking until the strawberries are very soft and the liquid has thickened slightly, about 8-10 minutes. The cornstarch will help it set properly once cooled. Pour the compote into a bowl and let it cool completely to room temperature, about 15-20 minutes, before using.

Once the cheesecake base from Step 3 is fully chilled, pour it into an ice cream maker and churn according to the manufacturer's instructions, usually 20-25 minutes, until it reaches a soft-serve consistency. Remove the chilled crust from the freezer and pour the churned ice cream directly over it, smoothing the top with a spatula. Freeze the assembled cake for at least 4 hours, or until the ice cream is completely firm.

Remove the cake from the freezer 5 minutes before serving to make it easier to slice. Run a thin knife around the inside edge of the pan and carefully release the springform sides. Pour or spread the cooled strawberry compote from Step 4 over the top of the cake. Arrange the fresh whole and sliced strawberries decoratively over the compote. I like to let the cake sit at room temperature for about 10 minutes before cutting—this makes slicing cleaner and the texture creamier while still keeping the ice cream frozen solid.