MAKE THE PASTRY CREAM: In a saucepan, combine ¾ cup of the milk with the sugar and basil; bring to a simmer. Remove from the heat and let stand for 15 minutes. Remove the basil; squeeze any milk back into the pan.
In a bowl, whisk the remaining ¼ cup of milk with the yolks and cornstarch. Slowly whisk the yolk mixture into the warm milk; bring to a simmer over moderate heat, whisking constantly until very thick, 2 minutes. Remove from the heat and whisk in the butter until melted. Scrape into a bowl. Press plastic wrap directly on the surface and refrigerate until chilled, 2 hours.
MAKE THE PASTRY: Preheat the oven to 375 degrees F. Spray a 14×4 1/2 –inch rectangular tart pan with a removable bottom with nonstick cooking spray. In a standing mixer, beat the hard-boiled egg yolk with the butter and confectioners’ sugar at medium speed until smooth, 2 minutes. Add the flour, potato starch and salt and beat at low speed until just combined. Using lightly floured hands, press the dough evenly over the bottom and up the sides of the tart pan. Refrigerate the crust for about 30 minutes, or until chilled. Bake the crust for about 25 minutes, until golden. Cool on a rack for 1 hour.
MAKE THE TOPPING: Increase the oven temperature to 450 degrees F. Line a baking sheet with parchment paper. Arrange the apricot halves cut side up on the paper and sprinkle all over with the sugar. Roast for about 20 minutes, until tender and lightly browned. Cool completely, about 30 minutes.
Unmold the crust onto a serving plate and spread the cream evenly in the crust. Arrange the apricots on the cream, cut side down, and brush with the melted jam. Cut the tart crosswise into strips and serve.