Brown the Butter: Melt ½ cup plus 1 tablespoon of unsalted butter in a medium saucepan over medium heat, stirring frequently until it deepens to golden brown. Let cool slightly.

Make the Crust: Combine 2 cups of crushed cookie crumbs, ½ cup of ground pecans, and ¼ cup of powdered sugar in a bowl. Add the cooled brown butter, mix well, and press into the springform pan. Bake at 350°F for 10 minutes, let cool.

Prepare the Filling: Beat 16 ounces of softened cream cheese and ¾ cup of brown sugar until smooth. Add ¼ teaspoon of fine salt and ¼ cup of brown butter. Mix in 1 cup of sour cream until combined.

Incorporate the Eggs: Add 3 large room-temperature eggs one at a time, mixing gently. Scrape the seeds from 1 vanilla bean pod into the filling and mix until combined.

Bake the Cheesecake: Pour the filling into the cooled crust and smooth the top. Bake at 325°F in a water bath for 1 to 1.5 hours until set around the edges.

Cool the Cheesecake: Turn off the oven and leave the cheesecake inside for 1 hour with the door slightly ajar. Refrigerate for at least 8 hours before serving.