Preheat oven to 425˚F (220˚C). Line two baking sheets with parchment or lightly grease.
In a large bowl, fold together the finely chopped hazelnuts, sugar, vanilla sugar, cinnamon, salt, rum, and egg white until evenly combined.
In a small bowl, beat together the egg yolk and milk to make the egg wash. Set aside.
On a lightly floured surface, roll the puff pastry into a 12 x 12 inch (30 x 30 centimeter) square.
Cut the puff pastry into 9 equal squares, each 4 x 4 inches (10 x 10 centimeters).
Place a strip of filling across the middle of a pastry square, leaving a 1 1/2 inch (4 centimeter) space clear on each side.
Brush the edge of one half of the square with beaten egg wash. Fold the square over, enclosing the nut mixture and sealing the edge.
Using a sharp knife, make slits about 1 inch (2.5 centimeters) deep and 1/2 inch (1.25 centimeters) apart along the long, sealed edge.
Gently pull the pastry side opposite the slits into a slight horseshoe shape to separate the slices, pulling apart any that are sticking together, to form a comb.
Transfer the pastry to the prepared baking sheet. Repeat with remaining puff pastry and filling.
Brush the tops with the remaining egg wash.
Bake in preheated oven until golden brown, 12-15 minutes. Transfer to a wire rack.
In a small bowl, whisk together the powdered sugar and lemon juice to form a smooth glaze. Drizzle over the pastries and allow to set briefly before serving.
If you don't want the glaze, you can simply dust the pastries with powdered sugar before serving.