Preheat oven to 375 degrees F. Bring a large pot of water to a boil.
Roll out the chilled puff pastry dough on a lightly floured surface to a 1/8 inch thickness. Cut the dough into six 7 inch squares. Put on a baking sheet and refrigerate while you prepare the peaches.
Combine cinnamon, 1/3 cup of sugar, brown sugar and flour. With a pastry blender, two knives or very light cool fingers, mix in the butter until it resembles coarse oatmeal.
Blanch the peaches for 10 seconds in rapidly boiling water. Remove quickly and drain well. Peel the peaches, cut in half, discarding pits. Toss the peach halves in lemon juice and then in the cinnamon mixture.
Put the remaining cinnamon mix into the center of half the peaches. Put the remaining peaches on top of the stuffed ones to reassemble as if whole.
Remove pastry from fridge. Put a peach in the center of each square of dough. Moisten the edges of the dough with cold water, then gently pull up the points of the square and pinch the seams together to seal.
Using a pastry brush, gently paint the dumplings with beaten egg. Sprinkle with the 2 teaspoons of sugar. Prick a few holes around the top to allow steam to escape.
Bake in preheated oven for about 40 minutes or until the peaches are tender and the pastry has browned.
When dumplings are done, remove from baking sheet and allow to cool for 20 minutes. These are delicious with vanilla or peach ice cream.
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