Whisk the flour, 1 tbsp sugar, and 1/2 tsp salt in a 3-quart / 2.8 L mixing bowl. Add the cold cubed butter and pinch it with your fingertips for about 3 minutes, flattening it into pea-size and almond-size flakes. Sprinkle in the ice water 1 tbsp at a time, tossing with a fork, until the dough clumps when squeezed. Press it into a 2.5 cm / 1-inch disk, wrap, and chill for 45 minutes.

While the dough chills, melt the 57 g butter in a 2-quart / 1.9 L light-colored saucepan over medium heat. Cook 6 to 8 minutes, swirling often, until the foam thins, the specks turn hazelnut brown, and the smell turns nutty. Pour it into a bowl, scraping in the browned bits, and cool for 10 minutes. You'll have about 3 tbsp.

Peel, core, and slice the apples 5 mm thick into a second 3-quart / 2.8 L bowl and toss with the lemon juice. Set a rack in the lower third of the oven and heat it to 200 C / 400 F.

Just before assembling, add the brown sugar, cornstarch, 1/2 tsp cinnamon, 1/4 tsp salt, vanilla, and 2 tbsp of the brown butter to the apples. Toss until no dry streaks remain and keep the remaining 1 tbsp butter warm for brushing.

Roll the dough on a lightly floured sheet of parchment into a 30 cm / 12-inch round about 3 mm thick. Slide the parchment onto a 33 x 46 cm / 13 x 18-inch rimmed sheet pan. If the dough turns soft and sticky, chill the pan for 10 minutes first, because warm dough lets butter leak out in the oven and leaves a greasy base.

Mound the apples in the center, leaving a 5 cm / 2-inch border and the bowl juices behind. Fold the border up and over the fruit in loose pleats. Brush the crust with the reserved warm brown butter, rewarming it for 10 seconds if it has set. Mix 2 tbsp sugar with 1/2 tsp cinnamon and sprinkle it over the crust.

Bake for 40 minutes, rotating the pan at the 20-minute mark, until the crust is deep golden brown, the juices bubble slowly near the center, and a paring knife slides easily into the apples.

Cool on the pan for 5 minutes, then slide the parchment onto a wire rack and cool 15 minutes more, 20 minutes total. Cut into 8 wedges and serve warm.