How to Make It
In a medium bowl, beat the butter and sugar until light and creamy. Beat in the egg and vanilla. Stir in the flour just until combined. Shape the dough into a ball and cut it in half. Flatten each piece into a 6-inch disk and wrap separately in plastic wrap. Refrigerate one half until chilled, for at least 1 hour or overnight; reserve the other half in the freezer for up to 1 month for future use.
On a lightly floured sheet of parchment paper, roll out the chilled dough to a 13-inch round; if the pastry becomes too soft, slide the paper onto a cookie sheet and refrigerate for 10 minutes. Invert the pastry onto an 11-inch tart pan with a removable bottom and peel off the paper. Press the pastry into the pan and trim the edges. Refrigerate the pastry for 30 minutes.
Preheat the oven to 350°. Line the tart shell with a large piece of foil and fill it with pie weights, dried beans, or rice. Bake the pastry on the bottom rack of the oven for 15 minutes. Remove the foil and weights and bake for about 15 minutes longer, or until the pastry is golden. Let the pastry cool in the tart pan. Leave the oven on.
Spread the hazelnuts on a baking sheet and toast for about 15 minutes, or until the skins are blistered. Rub the nuts together in a towel to remove as much of the skin as possible. Let the nuts cool, then coarsely chop them.
In a small saucepan, cook the butter over moderate heat until browned, about 1 minute. In a medium bowl, whisk together the eggs, egg yolk and sugar until light and fluffy. Whisk in the corn syrup, vinegar, browned butter, vanilla, and salt and stir in the hazelnuts. Pour the filling into the prebaked tart shell, spreading it evenly. Bake on the bottom rack of the oven for about 30 minutes, or until the filling is set. Let cool for at least 1 hour or up to 8 hours. Serve the tart warm or at room temperature with whipped cream.