Special equipment: a deep-frying thermometer
For the cheesecake mixture: Line a baking sheet with wax paper. Beat the cream cheese until smooth in a stand mixer fitted with the paddle attachment. Add the sugar, vanilla and graham cracker crumbs and beat until well combined. Scoop 1-tablespoon balls of the cheesecake mixture onto the prepared baking sheet. Freeze for about 30 minutes.
For the strawberry sauce: Meanwhile, combine the strawberries, sugar and lemon juice in a medium saucepan over medium-high heat. Cook, stirring often, until the sugar is dissolved and the berries begin to break down and to give off their juices, 5 to 7 minutes. Transfer to a blender or food processor and puree until smooth. Strain through a fine-mesh sieve to remove the seeds. Set aside.
For the batter: Heat 2 inches of vegetable oil in a deep, heavy-bottomed pot over medium heat until the oil registers 350 degrees F on a deep-frying thermometer. Line a baking sheet with paper towels and set aside.
In a medium bowl, whisk together the flour, sugar, baking powder and salt. Add the milk and whisk until smooth.
When the oil is ready, remove the cheesecake balls from the freezer. Working in batches, add the cheesecake balls to the batter a few at a time, turning to coat. Remove from the batter, allowing excess to drip off, and then carefully add to the oil. Fry, turning occasionally, until golden brown, 2 to 3 minutes per batch (allow the oil to return to temperature between batches). Remove from the oil with a slotted spoon and drain on the paper towels.
Serve warm, with strawberry sauce.