Sauté the finely diced apples in the butter, sugar, cinnamon, nutmeg, and lemon juice for 10 minutes. Then, add 2 tablespoons of spiced rum and continue to cook the apples until they are softened, roughly 5 minutes. Leave them to cool.
Preheat the oven to 350 F.
Cut out 6 squares of puff pastry and add 1 heaping tablespoon of the cooled apple filling to one corner of the pastry. Beat the egg and then brush a little around the edge of the pastry square, before folding the other side over the top to make a triangle. Crimp the edges using a fork, then brush with a little of the beaten egg to finish off. Complete all 6 pastries and then place them in the oven to cook for 20 to 25 minutes, until they are puffed up and golden.
In a small bowl, mix together the powdered sugar with the remaining tablespoon of spiced rum, the vanilla extract, and 1 tablespoon of water. Mix everything together until a smooth paste has formed, then drizzle it over the top of the cooled apple turnovers.