Make the crust: Pulse the flour, granulated sugar and salt in a food processor until combined. Add one-third of the butter and pulse until incorporated. Add the remaining butter and pulse until the mixture looks like coarse meal with bits of butter throughout, about 3 pulses. Add the vinegar and 3 tablespoons ice water; pulse 3 or 4 more times to combine. Squeeze the dough between your fingers; if it doesn't hold its shape, add 1 more tablespoon ice water and pulse again. Pat the dough into a disk, wrap in plastic wrap and refrigerate at least 1 hour. Position a rack in the lower third of the oven; preheat to 400 degrees F. Lightly butter a 9-inch tart pan with a removable bottom. Roll out the dough into a 10-inch circle on a floured surface. Ease the dough into the pan and press into the bottom and sides; trim the excess. Pierce the dough all over with a fork. Refrigerate until firm, about 30 minutes. Line the dough with aluminum foil and fill with dried beans. Bake until set, about 20 minutes. Remove the foil and beans; continue baking until golden, about 15 more minutes. Transfer to a rack and let cool completely.
Make the filling: Beat the goat cheese and 1/3 cup granulated sugar in a large bowl with a mixer on medium speed until creamy, 3 to 4 minutes. Add the 4 egg yolks and beat until combined, about 1 minute. Add the cornstarch, vanilla and cognac and beat until smooth, about 1 minute, scraping down the bowl as needed. In a separate large bowl, beat the 3 egg whites until foamy. Add the salt and the remaining 1 tablespoon granulated sugar and beat until soft peaks form, 1 to 2 minutes. Using a rubber spatula, fold the beaten egg whites into the yolk mixture until combined, then pour into the prepared crust. Bake until the filling is set and golden on top, about 20 minutes. Transfer to a rack and let cool completely.
Make the topping: Bring the granulated sugar, honey, cinnamon, cognac, vanilla and 1/3 cup water to a simmer in a small skillet over medium heat; cook until syrupy, 5 to 7 minutes. Add the apricots and cook 3 more minutes, then add the lemon juice. Let cool completely. Use a small knife to loosen the tart, then remove the springform ring. Dust with confectioners' sugar. Serve at room temperature with the apricot topping. Photograph by Johnny Miller