In a large skillet over medium heat, add the butter and brown sugar. Stir to combine and the sugar dissolves. Add the cinnamon and stir to combine. Add the apples, and coat in the butter mixture. Cook for 10-15 minutes, stirring occasionally until the apples are tender. Take off the heat and add in the vanilla, set aside to cool.
Preheat oven to 400 degrees. Pierce puff pastry sheets all over with a fork. Cut into 4ths. Add a small amount of the filling into the center of each square. Fold it over to form a triangle. Using a fork, poke a few holes on the top and use the fork to seal the edges together. (see tutorial video)
Brush with the egg mixture, bake for 15 minutes until golden brown.