Preheat oven to 200°C (400F) and line a large baking sheet with parchment paper.
Place diced apples, butter, brown sugar, cinnamon, nutmeg, and walnuts in a medium-sized saucepan on medium heat. Simmer for 5-7 minutes until apples are soft and syrup is released.
Dissolve 1 tsp. cornstarch in 1 tbsp. cold water and stir into apple mixture to thicken.
Transfer apple mixture to a glass bowl and cool completely.
Roll out puff pastry on a lightly floured surface and slice into squares each measuring around 12cm (4.5 inches).
Place 2-3 tsp. of cooled apple mixture (without too much liquid) in the middle of each piece of puff pastry. Dip one finger in some water and run it along the edges of each piece of pastry and then fold each square in half. Gently press the edges together using a fork.
To make egg wash, beat a small egg and whisk in 1 tbsp. of milk. Use a pastry brush to brush each turnover with egg wash, then sprinkle with some brown sugar.
Bake turnovers for 15 to 20 minutes until puff pastry is a deep golden brown. Allow to cool slightly (filling will be hot!), glaze as desired, and serve.
To make the glaze: stir in 1 tbsp. of milk with 5-6 tbsp. of powdered sugar. If glaze is too thick, add more milk. If it's too thin, add more sugar. Drizzle over cooled turnovers.