Preheat oven to 375º. Line a small baking sheet with parchment paper. Wrap the outside of a 10-inch round springform pan with foil.
Crust: In a small bowl combine the pecan halves, ½ teaspoon butter, ½ tablespoon sugar, brown sugar, cream and salt. Mix to combine and spread onto the prepared baking sheet. Bake for 16-18 minutes until the sugar is brown. Let cool until the sugar is no longer soft.
Reserve 12 of the pecans and add the rest to a food processor. Add graham crackers and process until fine. Add ½ cup of melted butter and ⅓ cup sugar and pulse until combined. Press the crumbs into the bottom and two-thirds up the sides of the springform pan and pat into place with the bottom of a measuring cup.
Filling: Clean out the food processor or use a stand mixer. Add the cream cheese and sugar and process until smooth. Add eggs, one at a time and process after each until they are incorporated. Add vanilla and sour cream and process until smooth. Pour into the crust. Bake for 55-60 mintues, or until the center is still jiggly and the edges have started to brown. Let cool completely on a wire rack then cover and refrigerate overnight.
Caramel Sauce: In a saucepan, add sugar, water and corn syrup and bring to a boil over medium-high heat. Cook until it turns a slightly yellow straw color, about 8 minutes, then reduce the heat to medium low and continue to cook until it turns an amber color, about 5 minutes. A thermometer should read 235º to 240º.
Remove from the heat and stir in cream, vanilla and salt, it will sizzle. Stir until it is smooth. Pour into a glass bowl and let it cool completely. Refrigerate until ready to use, up to two weeks. If it is too thick to pour, microwave at 10 second intervals.
Assemble: Remove the springform pan and place on a serving plate. Use a warm knife to cut into 12 slices, wiping down the knife between each slice. Place a reserved pecan on each slice and pour caramel over each slice.