Step 1: Preheat oven to 325°F (163°C). Prepare a 9-inch springform pan by lightly greasing it and wrapping the bottom and sides tightly with heavy-duty aluminum foil to prevent water from entering during a water bath.
Step 2: In a medium bowl, combine the graham cracker crumbs and 1/4 cup granulated sugar. Pour in 1/2 cup melted butter and mix until crumbs are evenly moistened.
Step 3: Press the crumb mixture firmly and evenly into the bottom of the prepared springform pan. Bake for 10 minutes, then remove from oven and let cool slightly.
Step 4: In a large mixing bowl, beat the softened cream cheese and 1 cup granulated sugar with an electric mixer on medium speed until smooth and creamy, scraping down the sides of the bowl as needed.
Step 5: Beat in the vanilla extract, then add the eggs one at a time, mixing on low speed just until combined after each addition. Do not overmix.
Step 6: Gently fold in the sour cream until the batter is smooth. Pour the cheesecake batter over the cooled crust in the springform pan.
Step 7: Place the foil-wrapped springform pan into a larger roasting pan. Pour hot water into the roasting pan until it comes about halfway up the sides of the springform pan.
Step 8: Carefully transfer the roasting pan to the preheated oven and bake for 60-75 minutes, or until the edges are set but the center still jiggles slightly when gently shaken.
Step 9: Turn off the oven, crack the oven door open, and let the cheesecake cool in the water bath inside the oven for 1 hour. Remove from the oven and water bath, unwrap foil, and let cool completely on a wire rack.
Step 10: While the cheesecake cools, prepare the pecan topping. In a medium saucepan, combine light corn syrup, brown sugar, and 4 tablespoons melted butter. Bring to a boil over medium heat, stirring constantly. Reduce heat and simmer for 1-2 minutes.
Step 11: Remove from heat and stir in the pecan halves until they are fully coated. Let the topping cool slightly, about 10-15 minutes, allowing it to thicken.
Step 12: Once the cheesecake is completely cool, pour the pecan topping evenly over the top. Cover the cheesecake loosely with plastic wrap and refrigerate for at least 6-8 hours, or preferably overnight, to allow it to set completely.
Step 13: To serve, carefully run a thin knife around the edge of the pan before releasing the springform sides. Slice with a hot, clean knife for best results.