Preheat the oven to 350°F (175°C) and combine graham cracker crumbs, sugar, and melted butter in a bowl. Press the mixture into a 9-inch springform pan and bake for 8-10 minutes until golden; then let it cool slightly and reduce the oven temperature to 325°F (160°C).

In a light-colored saucepan over medium heat, melt and cook ¾ cup unsalted butter for 5-7 minutes until it turns amber with nutty brown bits, then immediately transfer to a heatproof bowl to cool slightly.

Beat the softened cream cheese in a large bowl with an electric mixer until smooth and creamy for 2-3 minutes. Gradually add 1 ½ cups granulated sugar, beating until just combined and smooth, scraping the bowl as needed.

Pour the slightly cooled brown butter into the cream cheese mixture and beat on low speed until fully incorporated. Add the flour, vanilla extract, and scraped vanilla bean seeds, mixing until just combined.

Beat in the eggs one at a time on low speed until just incorporated, being careful not to overmix, then stir in the sour cream until combined to achieve a smooth and creamy filling.

Pour the cheesecake filling over the cooled crust in the springform pan. Wrap the bottom of the pan tightly with several layers of heavy-duty aluminum foil to prevent water from seeping in.

Place the foil-wrapped springform pan inside a larger roasting pan and carefully pour hot water into the roasting pan, ensuring it reaches about halfway up the sides of the springform pan.

Bake for 60-75 minutes, or until the edges are set and a 2-inch circle in the center still jiggles slightly when gently shaken.

Turn off the oven and prop the door open slightly, allowing the cheesecake to cool in the water bath inside the oven for 1 hour to prevent cracks.

Remove the cheesecake from the water bath, discard the foil, and transfer to a wire rack to cool completely at room temperature. Then, cover loosely and refrigerate for at least 6 hours, or preferably overnight, before serving.