Preheat oven to 350°F (175°C). Grease a 9-inch springform pan.
In a medium bowl, combine graham cracker crumbs, granulated sugar, and melted butter. Press mixture firmly into the bottom of the prepared springform pan to form the crust.
Bake the crust for 8-10 minutes. Remove from oven and let cool slightly.
In a large bowl, beat together cream cheese and granulated sugar until smooth. Beat in eggs one at a time, then stir in vanilla extract and sour cream.
In a small bowl, combine brown sugar and cinnamon. In another small bowl, cream together the softened butter for the swirl until smooth.
Pour about half of the cream cheese filling over the crust. Dollop spoonfuls of the cinnamon-sugar mixture and softened butter over the filling. Swirl gently with a knife or skewer to create a marbled effect.
Pour the remaining cream cheese filling over the swirl mixture. Gently tap the pan on the counter to remove any air bubbles.
Bake for 50-60 minutes, or until the edges are set and the center is slightly jiggly. Turn off the oven, crack the door open, and let the cheesecake cool in the oven for 1 hour.
Remove from oven and let cool completely on a wire rack. Then, refrigerate for at least 4 hours, or preferably overnight.
For the frosting: In a medium bowl, beat cream cheese until smooth. Gradually beat in powdered sugar, then stir in milk and vanilla extract until smooth and creamy.
Once chilled, remove the sides of the springform pan. Frost the top of the cheesecake. Slice and serve.
Notes
For best results, ensure all cheesecake ingredients are at room temperature.