Begin by preheating your oven to 400 degrees F. Line two baking sheets with parchment paper and set aside.
Next, in a large skillet, add the butter, brown sugar, and salt. Cook over medium heat until the butter melts. Add the apples and water, and stir to combine. Cover with a lid and cook for about 5 minutes. Then, remove the cover and cook for an additional 8 minutes or until the apples are tender. Stir as needed.
Remove from heat, stir in the cinnamon and vanilla, and allow to cool slightly. Once cooled, fold in 2/3 of the cream cheese (5 ounces), the flour, and egg yolk. Stir until evenly combined. (It's OK if there are some small cream cheese lumps.) Set aside.
On a lightly floured surface, roll out one pastry sheet into a 12-inch square. Cut the square into four 6-inch squares. Add 1/4 cup of the cream cheese and apple filling to the center of each pastry square. Then, brush the edges of each pastry square with water. Fold the pastry in half to form a triangle and seal the edges with your fingers or a fork. Repeat with the second sheet of puff pastry.
Transfer the pastries to your prepared baking sheets (four turnovers per sheet). In a small bowl, whisk together the egg and water and then brush this mixture over the top of each pastry. Bake for about 15 minutes or until the turnovers are beautifully puffed and golden brown.
While the turnovers cool, prepare the frosting. Using a hand mixer on medium speed, beat the remaining cream cheese until smooth. Slowly add the powdered sugar and the milk, one tablespoon at a time, to achieve your desired consistency. (I used less milk because I prefer thicker frosting for this recipe.)
Spread the frosting on the turnovers and serve. Leftover turnovers can be stored in an airtight container for a couple of days.