Step 1

For the Filling: In a large heatproof bowl, toss together apples, granulated sugar (use 1 cup if your apples are very tart), salt, vinegar, and spices. Let stand at room temperature, stirring occasionally, for at least 3 hours, or refrigerate (covered) up to 24 hours.

Step 2

Stir flour into macerated apple mixture, then set a colander over another bowl and drain, reserving liquid. Transfer liquid to a small saucepan; return apples to bowl. Bring liquid to a boil over medium-high heat, whisking constantly, until thickened to the consistency of loose pudding, 1 to 2 minutes. Stir back into apple mixture.

Step 3

For the Crust: Preheat oven to 400°F, with a rack in bottom third. On a lightly floured surface, roll out disk of dough 1/8 inch thick (about 13 inches in diameter). Fit it into a 9-inch pie dish. Trim overhang to 1 inch, fold edges under, and crimp as desired. Brush edges of crust with egg and fill with apple mixture; refrigerate for 15 minutes (to help crust hold its shape in the oven).

Step 4

For the Topping: In a medium bowl with an electric mixer, beat butter, brown sugar, baking powder, and salt until light and creamy, about 2 minutes. Add flour and oats and beat to combine.

Step 5

Crumble topping over filling and bake for 20 minutes. Reduce oven temperature to 375°F and bake until golden brown all over and bubbling in center, 50 to 60 minutes more. (If the top browns too quickly, tent it with foil.) Transfer to a wire rack and let cool completely, at least 6 hours.