In a saucepan, cook the apples, sugar, cinnamon, anise oil, and 2 tablespoons of water. Let cook over medium heat for about 10 minutes, stirring occasionally. Remove from heat and let cool. Remove the anise oil.
Roll out the puff pastry on a work surface. Cut out squares of puff pastry.
Preheat the oven to 385°F.
Place a few spoonfuls of apple puree in the center of the pastry squares. Fold the squares, folding one corner over the other. Seal the edges of the pastry with your fingers dipped in a little water.
Make a small incision in each pastry triangle.
In a small bowl, beat the egg and 1 tablespoon of water. Use a brush to coat each turnover with the beaten egg.
Place the turnovers on a baking sheet. Bake and let cook for about 20 minutes, or until the pastry is puffed and golden brown.