How to Make It
Preheat the oven to 350°. Spray an 11-inch fluted tart pan with a removable bottom with vegetable oil spray. In a food processor, pulse the flour with the sugar, baking powder and salt. Add the butter and egg yolk mixture and process just until the pastry comes together. Turn the pastry out onto a work surface and knead 2 or 3 times. Press the pastry evenly into the tart pan and refrigerate until chilled, at least 30 minutes.
Line the tart shell with parchment paper and fill loosely with pie weights. Bake for about 35 minutes, until the edge is set. Remove the parchment and pie weights and bake for about 20 minutes longer, until the tart shell is lightly browned all over.
In a very large skillet, melt 2 tablespoons of the butter. Add 1/2 cup of the sugar; cook over high heat until just brown. Add the apples, season lightly with salt and pepper and cook over moderate heat, turning once, until lightly caramelized, 12 minutes.
In a small saucepan, cook the remaining stick of butter over moderate heat until the milk solids brown and the butter is fragrant, about 7 minutes. In a medium bowl, using an electric mixer, beat the eggs with the remaining 1/2 cup of sugar until thick and fluffy, about 3 minutes. Beat in the flour and 1/4 teaspoon of salt, then beat in the browned butter, scraping the milk solids into the custard mixture.
Pour the custard into the shell; arrange the apples in a single layer of slightly overlapping circles. Bake in the lower third of the oven for 45 to 50 minutes, until the custard is puffed and richly browned. Transfer to a rack and cool before serving.