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Preheat the oven to 350 degrees F.
Coat a large saute pan with the butter and place over medium heat. Add the peaches, their juices and the lemon juice together and mix well. Then add the sugar, allspice, cinnamon, vanilla and nutmeg, and mix well. In a small bowl mix the corn starch with 1/4 cup water. Let the spices and peaches simmer for a few minutes, and add the corn starch mixture to thicken the peach juice, and simmer for 1 to 2 minutes more.
Lay one of the puff pastry sheets out and sprinkle the flour on top. Place one of the serving dishes on top of the puff pastry and cut around the dish (cutting a little wider than the dish, as this will be the top crust). Repeat this for the remaining servings.
Cut the remaining puff pastry into bite-size pieces and place on a baking tray and partially bake the pieces, about 10 minutes, then remove from the oven and set aside to cool.
Place the pieces of partially-baked puff pastry at the bottom of the dishes to create a bottom crust.
Scoop the peaches into each dish and fill to the top. Cover each with a cut piece of puff pastry and pinch the sides to seal (if the dough is too big, feel free to trim down to fit the size of the dish a bit better).