Preheat oven to 400ºF (200ºC). Mix sugar with cornstarch. In a large bowl, stir together peaches, blackberries, basil and lemon juice; add sugar mixture. Gently stir to combine and coat fruit well.
Lay pastry on a flat, lightly floured surface; divide fruit mixture evenly among pastry squares.
Lightly brush corners of pastry squares with some of the egg. Seal pastries by pinching together opposite corners. Transfer pastries to a baking sheet lined with parchment paper; gently brush with some of the egg. Bake until golden, 17 to 20 min.