Special equipment: A 9-inch springform pan
For the crust: Position a rack in the center of the oven, and preheat to 325 degrees F. Butter the inside of the springform pan.
Pulse the graham crackers in a food processor until finely ground. Add the butter, sugar and salt, and pulse until the mixture comes together and holds its shape when squeezed. Pour the mixture into the prepared pan, and press it into the bottom to make an even layer. Bake until the crust is just firm, about 10 minutes. Remove and let cool completely. Place the pan on a large piece of foil, and fold the foil up the sides of the pan. (This helps seal the pan, to keep water out while baking.)
For the filling: Wipe out the food processor. Pulse the cream cheese, sugar and sour cream until completely smooth, without any lumps, scraping down the sides of the bowl a few times. Add the eggs and vanilla, and pulse until incorporated. Pour the filling into the crust, and set the pan inside a roasting pan. Fill the roasting pan with enough hot water to go halfway up the sides of the springform pan (but not above the foil).
Bake the cheesecake until the edge is set but the center jiggles slightly (when you gently shake the pan, the cheesecake should move in a cohesive wave motion), 1 hour 10 minutes to 1 hour 20 minutes. Turn the oven off, and open the door for a few seconds to let out some heat. Close the door, and let the cheesecake sit in the oven until cooked completely through, 1 hour more.
Transfer the springform pan to a cooling rack. Run a knife around the edge, and let the cheesecake cool to room temperature. Wrap and refrigerate overnight or up to 2 days.
For the topping: Preheat the oven to 350 degrees F. Spread the pecans out on a baking sheet, and bake until they turn a few shades darker and are nicely toasted, about 15 minutes. Let cool, then halve each pecan lengthwise.
Meanwhile, heat the cream, butter, corn syrup, vanilla and salt in a small saucepan over medium heat until just steaming. Reduce the heat to low to keep the cream mixture warm.
Cook the sugar in an even layer in a medium skillet over medium heat, without stirring, until it begins to melt. Gently swirl the pan as needed as the sugar continues to melt and begins to caramelize and turn lightly golden in spots. Keep cooking and swirling until the caramel is a uniform deep amber. Remove from the heat, and slowly whisk in the warm cream mixture (watch out for ferocious bubbles). Return the caramel sauce to medium heat, and cook to reduce slightly, about 2 minutes. Let cool. To see if the sauce is ready for topping the cheesecake, dip a spoon into it and let the sauce drop off; it's ready if it easily and slowly drips off the spoon. (The sauce can be refrigerated up to 2 days; reheat in the microwave.)
To assemble: Spoon the sauce into the center of the cheesecake, and let it ooze to the edge. If needed, swirl the sauce with the back of a spoon to help it along. Sprinkle the pecans in a generous layer over the sauce. The cheesecake can now be served or refrigerated for up to 4 hours.
To slice the cake, dip a sharp, heavy knife into warm water, wipe the knife dry and slice into the cake. Wipe the knife clean and repeat.
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