Combine the peaches, water, vanilla seeds and pod, granulated sugar, brown sugar, lemon juice and salt in a saucepan. Bring to a simmer and cook until syrupy and the peaches are soft and just beginning to break down, 12 to 18 minutes. (Add more water if the mixture is too thick.) Discard the vanilla pod and mash the peaches; let cool.
Preheat the oven to 425˚ F. Line a baking sheet with parchment paper. Gather the pie dough into 2 small compact squares; roll out each into a 12-inch square. Cut each in half, then cut each half into thirds to make twelve 4-by-6-inch rectangles.
Arrange 6 rectangles on the baking sheet and top each with 3 tablespoons of the peach compote. Cover each with another rectangle of dough; crimp the edges with a fork to seal. Cut a small vent into the center of each top; brush with the beaten egg. Bake until golden brown, 20 to 25 minutes.