Preheat your oven to 350°F (175°C). In a mixing bowl, combine graham cracker crumbs, finely ground pecans, brown sugar, and melted butter until the mixture resembles wet sand. Press this mixture evenly into the bottom of a greased 9-inch springform pan. Bake for 10 minutes until lightly golden, then remove from oven and let cool.

Lower the oven temperature to 300°F (149°C). In a large mixing bowl, beat the softened cream cheese until smooth and creamy. Gradually add granulated sugar, sour cream, vanilla extract, and salt, mixing between additions. Gently incorporate the lightly beaten eggs until just combined.

Pour the cheesecake filling into the cooled crust, spreading it evenly. Place the springform pan in the preheated oven, and bake for 40 minutes. The cheesecake should be set around the edges but still slightly jiggly in the center. Allow it to cool.

While the cheesecake bakes, create the pecan topping by combining brown sugar, dark corn syrup, dark rum, and unsalted butter in a saucepan over medium heat. Stir until it comes to a gentle boil, then continue cooking for about 2 minutes. Remove from heat and let it cool slightly.

Whisk together the eggs, salt, and heavy cream in a separate bowl. Slowly whisk the cooled syrup mixture into the egg mixture, stirring in the coarsely chopped pecans.

Spoon the pecan topping over the partially baked cheesecake, spreading it gently. Return the cheesecake to the oven for an additional 40 to 50 minutes, monitoring closely to prevent overcooking.

Once the cheesecake is perfectly baked, allow it to cool completely in the springform pan. Refrigerate for at least 4 hours, preferably overnight, to achieve a silky texture.