Melt butter in a light-colored saucepan over medium heat, stirring until it foams, turns golden brown, and nutty bits form at the bottom (5-8 minutes). Immediately pour into a heatproof bowl to cool slightly and stop the cooking process.

Preheat oven to 350°F (175°C). Combine graham cracker crumbs, sugar, and ⅓ cup cooled brown butter, then press into a 9-inch springform pan. Bake for 8-10 minutes, cool slightly, and reduce oven to 325°F (160°C).

Wrap the springform pan tightly with several layers of heavy-duty aluminum foil to prevent water seepage. Bring a kettle of water to a boil.

In a large bowl, beat softened cream cheese with an electric mixer until smooth (2-3 minutes), then scrape down the sides of the bowl.

Gradually add granulated sugar to the cream cheese, beating until smooth, then mix in the remaining ½ cup of cooled brown butter until incorporated.

Add eggs one at a time, beating on low speed until just combined after each, then mix in the egg yolk without overmixing.

Add heavy cream, vanilla bean seeds (or paste/extract), and all-purpose flour, mixing on low speed until smooth. Finally, stir in the sour cream (or Greek yogurt) until fully incorporated.

Pour the filling over the cooled crust in the prepared springform pan. Place the springform pan into a larger roasting pan, then carefully pour boiling water into the roasting pan until it reaches halfway up the sides of the springform.

Bake for 60-75 minutes, until the edges are set and the center jiggles slightly.

Turn off the oven and leave the cheesecake inside with the door ajar for 1 hour to cool slowly and prevent cracks. Remove from oven, carefully take the springform from the water bath, remove foil, and cool completely on a wire rack.

Once completely cool, cover loosely with plastic wrap and refrigerate for at least 6 hours, or overnight, before serving.