Preheat oven to 400°F. Line a baking sheet with parchment paper.
In a skillet, melt butter over medium heat. Add diced apples, brown sugar, cinnamon, and nutmeg. Cook until apples are just tender, about 5–6 minutes.
In a small bowl, whisk the cornstarch and 1 tablespoon of water to create a slurry. Stir the slurry into the apples in the skillet and continue to cook for 1–2 minutes, stirring, until the filling has visibly thickened. Remove from heat and let cool completely.
On a lightly floured surface, unfold each sheet of puff pastry. Roll gently to smooth creases. Cut each sheet into 4 squares (8 total).
Place a spoonful of cooled apple filling in the center of each square. Fold into a triangle and press edges to seal. Crimp edges with a fork.
Place turnovers on prepared baking sheet. Beat egg with 1 tablespoon water, then brush over each turnover. Cut a small slit on top of each to allow steam to escape.
Bake 20–25 minutes, until golden brown and puffed. Cool slightly.
For the icing, whisk together powdered sugar, milk, and vanilla until smooth. Drizzle over cooled turnovers before serving.