1 Roll out the pastry: Set the oven to 375F. On a lightly floured counter, roll the pastry dough to a 12-inch round, giving it a quarter-turn after every few rolls so it does not stick to the counter.
Lift the pastry onto the rolling pin and ease it into the tart pan. With floured fingertips, press the pastry into the corners and up the sides of the pan. With scissors or kitchen shears, trim the top edge of the pastry so it comes about 1/2-inch above the pan. Tuck the top 1/2-inch of the pastry onto itself to make a hem and crimp.
2 Chill the pastry crust: Set the tart pan on a rimmed baking sheet and freeze for 15 minutes or refrigerate for at least 30 minutes, or until firm and cold (but not frozen).
3 Blind bake the crust: Prick the chilled dough all over. Lay a piece of foil over the tart, shiny side down, and press it into the edges of the pan. Fill the foil with pie weights or dried beans. Bake the pastry for 15 minutes, or until the edges start to brown. Carefully lift out the foil and pie weights or beans. Continue baking the pastry for 5 minutes more, or until the surface starts to look baked.
4 Set the tart crust aside to cool. Leave the oven on.
5 Toast the nuts: Spread the nuts in a baking dish or on a baking sheet. Toast them in the oven, stirring several times, for 8 minutes or until they are aromatic. Leave the oven on.
6 Begin making the filling: In a large saucepan, melt the butter over low heat just until it coats the bottom of the pan. Do not let the butter brown. Add the granulated sugar and brown sugar. Cook, stirring, until the sugar dissolves and combines with the butter in a smooth mixture. Add the chocolate and cook, stirring, until it melts. Remove the pan from the heat and let the mixture cool to lukewarm.
7 Finish the filling: Stir the eggs into the lukewarm chocolate mixture one by one, followed by the corn syrup and vanilla.
8 Assemble the tart: Leave the tart on the baking sheet used in the chilling step. Spread the toasted pecans an even layer in the bottom of the tart shell. Pour the chocolate filling over top and spread to the edges of the crust.
9 Bake the tart: Bake for 25 minutes, or until the filling is set in the middle and puffs slightly, and the pastry is browned at the edges. If the pastry seems like it's becoming too brown, cover the edges loosely with foil. Set on a rack to cool.
10 Unmold the tart and serve: Set the cooled tart on a bowl so the outer ring falls off. Carefully use 2 wide metal spatulas to lift the tart from the bottom of the pan and transfer to a large platter. Sprinkle with powdered sugar and cut into slices.
If serving the next day, cover the tart loosely with foil and refrigerate. Let it come to room temperature before serving.