In a food processor, pulse 1 1/4 cups flour, 1/2 tsp salt, and 1 tbsp sugar. Add 1/2 cup cubed cold butter and pulse until pea-sized crumbs form. Drizzle in 3 tbsp ice water, pulse until dough holds together, adding up to 1 more tbsp water if needed. Shape into a disk, wrap, and chill 30 minutes.

Toss the apple slices with 1/2 cup sugar, 2 tbsp flour, 1 tsp cinnamon, nutmeg, pinch salt, and lemon juice until evenly coated.

Roll the chilled dough into a 12-inch circle, ease it into a 9-inch pie plate, trim, fold under the edges, and crimp. Fill with the apple mixture.

Preheat oven to 375°F (190°C). Mix 3/4 cup flour, oats, brown sugar, and 1/2 tsp cinnamon, then work in 6 tbsp cold butter with your fingers until crumbly. Sprinkle over the apples. Brush the crust edge with egg wash if using.

Bake on a foil-lined sheet in the lower third of the oven for 50 to 60 minutes, until the crumble is deep golden and filling bubbles. Cool at least 1 hour, then slice and serve.