Combine the cinnamon, 1/3 cup of sugar, brown sugar and flour. With a pastry blender mix in the butter until it resembles coarse meal.
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. Place on a baking sheet and refrigerate.
Blanch the peaches for 10 seconds in rapidly boiling water. Remove and drain. Peel the peaches, split in half, and remove the pits. Toss the halves in lemon juice and then in the cinnamon mixture. Reassemble the peach halves with the remaining cinnamon mixture in the cavity.
Remove baking sheet from refrigerator. Place a peach in the center of each square of dough. Paint the edges of the dough with cold water. Bring the points of the square together at the top and pinch the seams together to seal.
Paint the dumplings with the beaten egg. Sprinkle with the 2 teaspoons of sugar (use crystal sugar for these two tablespoons if you have it available). Prick a few holes around the top to allow steam to escape. Bake for about 40 minutes or until the peaches are tender and the pastry has browned. When dumplings are done, remove from the baking sheet and allow to cool for 20 minutes.
To serve, spoon some of the sauce onto each serving plate, place dumpling in the center and drizzle some additional sauce over the top. You may serve with vanilla or peach ice cream or peach sorbet (recipe follows).