Make first pastry disk as directed through step 5 of Flaky All-Butter Pastry recipe in a regular 9-in. pie pan. Roll second disk into a 14-in. round; cut into 1-in.-wide strips, using a fluted pastry cutter or a knife. Transfer strips to a baking sheet; chill along with pie shell until needed.
Peel, core, and cut apples, half into 3/4-in.-thick slices and half into 1-in. chunks; you should have 7 to 8 cups. Toss with 1/3 cup sugar and the lemon juice.
Melt butter in a 12-in. frying pan over medium heat. Add apples. Cook, turning gently with a spatula, until apples soften at edges but are still crisp in the middle, 15 minutes. Remove from heat and add quinces. Transfer to a rimmed baking sheet, let cool, then chill until cold.
Preheat oven to 375°. Evenly fill pie shell with fruit and juices. Weave pastry strips over pie in a lattice: Evenly space half the pastry strips over pie in one direction (an odd number is best). Starting at center, fold back every other pastry strip and lay a strip across the rest. Unfold first set of strips so they're flat. Fold back second set of alternating pastry strips and lay another strip across pie. Unfold strips flat. Repeat on other side.
When finished, trim ends of strips even with bottom crust and tuck edges under crust so flush with and sitting on top of rim; you can leave the rim plain or flute it. Sprinkle remaining 2 tsp. sugar over pie and wrap pie rim with 4 strips of foil.
Bake 20 minutes. Reduce heat to 350° and bake 40 minutes more, then remove foil. Continue baking until crust is golden brown and juices are bubbling in center, 20 to 30 more minutes. Let cool on a rack at least 2 1/2 hours.
Make ahead: Up to 1 day.