In a small saucepan, combine the diced apples, granulated sugar, brown sugar, cinnamon, and nutmeg. Cook over medium heat, stirring occasionally, for 5-7 minutes, or until apples begin to soften.

In a small bowl, whisk together the cornstarch and water until smooth. Add this slurry to the apple mixture in the saucepan, along with the butter. Continue to cook, stirring constantly, for 1-2 minutes, until the filling thickens. Remove from heat and let cool slightly while you prepare the biscuits.

Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper.

Separate the biscuit dough. On a lightly floured surface, gently flatten each biscuit into an oval shape, about 4-5 inches long.

Place about 1-2 tablespoons of the cooled apple filling on one half of each flattened biscuit, leaving a small border.

Fold the other half of the biscuit dough over the filling to create a turnover shape. Press the edges firmly with your fingers, then use the tines of a fork to crimp and seal the edges, ensuring the filling is enclosed.

Place the assembled turnovers on the prepared baking sheet. Brush the tops of each turnover with the beaten egg wash.

Bake for 12-15 minutes, or until the turnovers are golden brown and puffed.

While the turnovers are cooling slightly, in a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth and creamy. If the glaze is too thick, add a few more drops of milk; if too thin, add a little more powdered sugar.

Drizzle the glaze generously over the warm turnovers. Serve immediately.