Preheat oven to 400°F.
Onto a flat work surface lightly dusted with flour, unfold one sheet of puffed pastry. Lightly dust the surface of the pastry with flour. With a rolling pin, lightly roll until square spreads to the size of a 12" x 12" square. Slice into four evenly sized squares.
Remove skins from peaches, slice in half and remove pits. Set aside.
In a small mixing bowl, combine brown sugar, sugar, butter, vanilla, cinnamon and nutmeg. The mixture will be crumbly.
Into one half of each peach, spoon 1 tbs. of the sugar & spices mixture into where the peach pit once was. Sandwich with other half of peach.
Place peach onto one puffed pastry square. Gently fold each corner to the top-center of each peach. Repeat with each peach.
In a small bowl, combine sugar and cinnamon.
With a pastry brush, brush melted butter over each pastry-covered peach. Then dust each with sugar. Sprinkle cinnamon-sugar over each peach.
Bake for 30 - 40 minutes or until lightly browned and crispy.