Place the potatoes in a pot with enough water to cover. Bring to a boil over medium-high heat, and cook until tender. Remove from heat, drain, and put through a ricer or mash.
Place the riced potatoes onto a large clean work surface. Crack the egg over the top, then gradually work in the flour with your hands until you get a nice stiff dough. You may not need to use all of the flour, or you may need to use more. This could take as long as 30 minutes, or just seem like it. Let the dough rest for a minute before rolling.
On a lightly floured surface, roll the dough out to 1/4 inch thickness. You may need to roll out 1/2 at a time. cut the dough into 8x8 inch squares - larger or smaller depending on the size of your peaches. Wrap each peach in a square of dough, and pinch all of the seams to seal it in.
Bring a large pot of water to a boil. Place the peach dumplings into the water. The peach should be as close to covered with the water as it can get. We usually had 2 or 3 pots with boiling water because you can only fit about 3 to 4 peaches per pot. Boil for 20 minutes, rotating after 10 for even cooking. Remove from water using tongs or a large slotted spoon.
To serve, place a peach dumpling onto a plate, and cut it up. Remove the pit, and sprinkle with melted butter, sugar and vanilla wafer crumbs.
