Preheat oven to 375 degrees F. Line the bottom of a 9-inch springform pan with parchment paper and clip the sides in around it; set aside.
In a medium bowl, combine graham-cracker crumbs and sugar, then make a crater in the crumb mixture and pour in the melted butter. Stir until evenly coated. Add crumb mixture to the prepared pan and press into the bottom and about halfway up the sides of the pan.
Bake for 10 minutes, then reduce oven heat to 325.
While crust is baking, add 1 1/4 cups pure maple syrup to a small saucepan, bring to a boil, and cook until syrup has been reduced to a cup, stirring occasionally. Cool to room temperature.
In a large bowl or stand mixer fitted with a paddle attachment, cream together the cream cheese and brown sugar until smooth. Beat in sour cream, flour, vanilla, and salt. Slowly beat in the cooled syrup. Add in the beaten eggs and mix on low speed just until combined.
Pour into the prepared crust and smooth out so the batter is even.
Place the springform pan in a larger baking pan, and fill larger pan with 1 inch of hot water to create water bath.
Bake at 325ºF for 1 1/2 hours or until center has set and the top has lost its sheen. Remove from water bath and cool on a wire rack for 10 minutes, then use a paring knife to run along the edge of the cake to separate it from the pan.
Cool for another hour on wire rack. Transfer to fridge to cool for an addition 6 hours, preferably overnight.
To make the topping just before serving, line a lipped baking sheet with parchment paper and preheat oven to 450.
Slice apples into 1/8-inch-thick slices and then slice in half; arrange apples on the pan. Use a basting brush to brush apples with 1/4 cup maple syrup. Sprinkle with brown sugar and apple pie spice, then bake for 15 minutes.
Remove apples from oven and use the basting brush to brush the apples with the juice from the pan; broil on low for 3 to 5 minutes.
Remove pan sides from cheesecake and arrange apples on it. Finish by drizzling with remaining 1/4 cup maple syrup. Slice with a sharp knife.