Preheat the oven to 450 degrees. On a lightly floured surface, cut the puff pastry into 6 equal portions. Roll each portion into 6x6-inch square. Cut each cheese log into 3 equal portions.
Arrange each cheese portion on a pastry square. Brush the edges of the pastry with the egg. Draw the sides of the pastry up to form a pyramid. Press the edges together halfway up the sides, leaving the center open. Tuck a sprig of basil into the center of each tart.
Arrange the tarts on a baking sheet. Bake 15 minutes or until the cheese softens and the pastry is golden brown and puffed. Serve warm.