Preheat the oven to 375 degrees F. Melt the butter in a large saute pan over medium heat, tilting the pan to coat. Add the peaches, reserved peach juice and lemon juice and mix well. Add the sugar, allspice, cinnamon and nutmeg and mix well. Let the peaches simmer until the sugar dissolves, about 5 minutes.
Combine the cornstarch with 1/4 cup water in a small bowl, stirring to combine. Stir the mixture into the peaches and allow to simmer for a minute or two to thicken the liquid. Stir in the vanilla extract.
Lay the puff pastry sheets out on a work surface and sprinkle a pinch of flour or cornstarch on top. Lay the ramekins on top of the puff pastry and cut around the dishes a little bigger than each dish. Cut the leftover puff pastry into bite-sized pieces and place on a baking sheet. (Reserve the pastry rounds.) Bake the bite-sized pastry pieces until partially baked, about 10 minutes. Set aside to cool slightly.
Place a few of the bite-size puff pastry pieces in the bottom of each ramekin to create a bottom crust. Spoon the peaches into the ramekins, filling to the top. Top each dish with a puff pastry round, pinching the sides to seal. If the dough is too big, feel free to trim it to fit the dishes better. Gently brush the dough with the beaten egg and sprinkle with sugar and cinnamon. Place the cobblers on a baking sheet and bake until the puff pastry is golden brown, 12 to 15 minutes.