In a 12-inch skillet, melt the unsalted butter over medium heat. Add the apples and toss gently to coat in the melted butter. Add the lemon juice and brown sugar and stir until the apples are well coated.

Add the cinnamon, ginger, nutmeg, and salt and continue to cook, stirring occasionally, until the sugar has melted, and the mixture has become saucy. 

In a small bowl, whisk the sugar and flour to combine. Sprinkle this mixture over the surface of the pot and immediately stir to combine. Let this mixture cool completely before continuing.

When the filling is cool, preheat the oven to 400°F/205°C with the oven rack in the center. 

On a lightly floured surface, roll out the pie dough to ¼ in/6 mm thick. Cut the dough into 1-inch-wide strips vertically, then cut into 1-inch strips horizontally, making lots of 1-inch-wide squares of dough.

In a medium bowl, whisk the egg and water well to combine. Add the pie crust and toss well until each piece is
coated in egg. Transfer the crust squares to the surface of the pie, aiming to apply in a fairly even layer (but allowing some of the apple filling to still peek through).

Garnish the surface generously with turbinado sugar, and transfer to the oven. Bake until the crust is deeply golden brown, and the filling is bubbly, 30-35 minutes. Cool 10-15 minutes, then serve warm with Tillamook Vanilla Bean Ice Cream.