Preheat your oven to 325°F (163°C). In a medium bowl, combine graham cracker crumbs, melted butter, and brown sugar until well mixed. Press this mixture firmly into the bottom of a 9-inch springform pan, ensuring it's even. Bake for 10 minutes until slightly golden, then remove and let it cool.

In a saucepan over medium heat, melt the unsalted butter, stirring frequently to avoid burning. Continue cooking until the butter turns golden brown and emits a nutty fragrance, about 5-7 minutes. Allow it to cool slightly.

Using an electric mixer, beat the softened cream cheese and granulated sugar in a large bowl until creamy and smooth, about 2-3 minutes. Add the vanilla bean paste and mix in well.

Add the large eggs one at a time, mixing just until combined. Gently stir in the sour cream and the cooled brown butter until fully blended. Avoid overmixing to keep the filling silky.

Pour the filling over the cooled crust, smoothing the top with a spatula. Bake in the preheated oven for 55-65 minutes until edges are set and the center is slightly jiggly.

Turn off the oven and let the cheesecake cool inside for an additional hour. Cool completely at room temperature before refrigerating for at least 4 hours, or overnight.

Run a knife around the edges of the springform pan to release the cheesecake. Remove the sides and slice into pieces. Serve with fresh berries, caramel sauce, or whipped cream.