Make the Tart Shell: In a food processor, pulse the flour with the sugar and salt. Add the butter and pulse until the mixture resembles a coarse meal. Add the ice water and vanilla extract and pulse just until the dough comes together. Transfer the dough to a lightly floured work surface; flatten it into a disk. Wrap the disk in plastic and refrigerate for 1 hour, until firm.
Preheat the oven to 350°. On a lightly floured work surface, roll out the dough to a 15-inch round, 1/4 inch thick. Transfer the round to a 12-inch fluted tart pan with a removable bottom; gently press it over the bottom and up the side. Trim any excess. Refrigerate the tart shell for at least 20 minutes, until firm.
Line the tart shell with foil and fill with dried beans or pie weights. Bake for about 50 minutes, until the shell starts to brown around the edges. Remove the foil and weights and bake for about 25 minutes longer, until the shell is cooked through. Transfer to a rack and let cool. Lower the oven temperature to 325°.
Make the Filling: Spread the slivered almonds on a large rimmed baking sheet. Toast in the oven for about 6 minutes, until lightly browned. Let cool.
Meanwhile, in a medium nonreactive saucepan, bring the wine to a boil. Add the apricots, cover and simmer over moderate heat until plumped, about 10 minutes. Remove from the heat.
In a small skillet, cook the butter with the vanilla bean seeds over moderate heat until browned, about 4 minutes.
In a food processor, pulse the toasted almonds with the confectioners' sugar, flour and salt until finely ground. Add the eggs and pulse until just combined. Add the browned butter and the almond extract and pulse until smooth.
Drain the apricots and pat dry. Pour the almond filling into the tart shell. Nestle the apricots into the filling in concentric circles. Bake the tart for about 50 minutes, until the filling is golden brown and set. Transfer to a rack to cool. Cut into wedges and serve warm or at room temperature, dolloped with sweetened whipped cream.