For the sticky pecan topping: In a medium saucepan, combine the butter, brown sugar, heavy cream, maple syrup and salt and set over low heat until the butter is melted. Increase the heat, bring to a full boil and boil for 5 to 6 minutes; be careful to not let it boil over. Remove from the heat, add the vanilla and stir in the nuts. Let cool to room temperature, then chill overnight. This will help thicken up the caramel.
For the cinnamon filling: Combine the brown sugar, cinnamon, salt and cayenne if using. Set aside.
For the cookies: Unfold each piece of puff pastry on a lightly floured work surface. Roll the pastry ever so slightly to 10-by-10 inches. Brush off any excess flour from each side.
Divide the cinnamon sugar filling between the two squares of dough, spreading it to three of the edges and leaving 1/4 inch at the top edge (for sealing the dough). Lightly roll over the sugar to ensure an even layer. Begin rolling up the pastry from the edge closest to you, toward the top of the dough. Make sure the roll is very tight, especially the first part-it needs to be compact. Paint the top edge with the egg white to seal the roll. Once sealed, gently roll the log to make it perfectly round. The diameter should be 1 1/2 to 1 3/4 inches. Freeze the rolls until firm, 1 hour or overnight. (Properly wrapped, the rolls can be kept frozen for 2 to 3 weeks.)
Preheat the oven to 375 degrees F. You'll need three sheet pans lined with parchment and a cooling rack that fits the pan. Remove both logs (or bake only one) from the freezer. Cut each log into 1/2-inch disks. You should get 20 to 21 slices from each roll. Arrange the slices on the prepared sheet pans spaced 2 inches apart (they will spread). Lay a sheet of parchment over the cookies and place the cooling rack (feet side up) on top. We want the contact to force the cookies to bake flat; we're denying the puff pastry to puff.
Bake one sheet pan of cookies at a time (keep the other sheet pans in the refrigerator until baking). Check after 10 minutes: The cookies should be beginning to evenly brown on the tops and bottoms. Rotate the pan and continue to bake until golden brown, another 5 to 6 minutes. Remove from the oven and carefully lift off the rack and the parchment. Let cool before removing the cookies from the pan. Continue baking the remaining cookies. Trim the sugar from around the edges.
Once the cookies are cool, arrange them on a cooling rack with a parchment-lined sheet pan underneath to catch drips. Remove the pecan topping from the refrigerator and stir until well incorporated. Place 1 small spoonful of the pecan topping on top of each cookie. These are best eaten right away.