In a medium bowl, mix together the graham cracker crumbs, melted unsalted butter, and brown sugar until well combined. Press this mixture evenly into the bottom of a 9-inch springform pan. Preheat your oven to 325°F (163°C) and bake the crust for about 10 minutes until lightly golden. Let it cool completely.

Melt the unsalted butter in a medium saucepan over medium heat. Stir continuously until the butter turns a golden brown color and releases a nutty aroma, which should take around 5-7 minutes. Remove from heat and let it cool slightly.

In a large mixing bowl, use an electric mixer to beat the softened cream cheese until smooth and creamy, about 2-3 minutes. Gradually add granulated sugar and continue mixing until fully incorporated. Add the vanilla bean paste or seeds and mix just until combined.

Add the eggs to the cream cheese mixture one at a time, beating gently after each addition until just combined. Carefully incorporate the sour cream and the cooled brown butter until everything is well blended.

Pour the filling into the cooled crust and smooth the top with a spatula. Place the springform pan in your preheated oven and bake for 55-65 minutes. The edges should be set, while the center still jiggles slightly when shaken.

Once baked, turn off the oven and leave the cheesecake inside for 1 hour to cool gradually. After an hour, carefully remove the cheesecake and allow it to cool completely at room temperature. Wrap it tightly in plastic wrap and refrigerate for at least 4 hours.