Preheat the oven to 350°F. Spray an 11-inch fluted tart pan with a removable bottom with cooking spray (can also use 8x11-inch tart pan). 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. Remove the pastry dough to a lightly floured work surface and knead 2 or 3 times. Press the dough evenly into the pan and refrigerate until chilled, at least 30 minutes.
Line the tart shell with parchment paper and fill loosely with pie weights (or beans). Bake for about 20 minutes, until the edge is set. Remove the parchment and weights and bake for 10 to 20 minutes longer, until the tart shell is lightly browned all over.
PREPARE THE FILLING:
While shell is baking, prepare the filling. In a large skillet, melt 2 tablespoons of the butter. Add ½ cup of the sugar; cook over high heat until lightly browned. Add the apples, season lightly with salt and pepper and cook over moderate heat, turning once or twice, until caramelized, 10 to 12 minutes.
In a small saucepan, cook the remaining stick of butter over moderate heat until milk solids brown and butter is fragrant, about 7 minutes (stirring often). In a medium bowl, use an electric mixer to beat the eggs with 1/2 cup sugar- until thick and fluffy (about 3 minutes). Beat in the flour and 1/4 teaspoon salt, then beat in the browned butter, scraping the milk solids into the custard mixture.
Pour the custard into the shell; arrange apples in a single layer, slightly overlapping in a decorative pattern. Bake in the center of the oven for 35 to 45 minutes, or until custard is puffed and richly browned. Transfer to a rack to cool before serving. Serve with a dollop of whipped cream or a scoop of vanilla ice cream, if desired.
Notes
*I find that many of the apples sink into the custard and disappear from sight, so I save a few apple slices, then bake the tart for about 15 minutes, then open the oven and add a few more apple slices along the top (when the custard begins to set but is still soft)- it works perfectly.
*I make this tart the night before, covered it with plastic wrap and refrigerate it. The next day, it's easy to slice with a bread knife when cold and firm, and then it can be served at room temperature