In a saucepan, combine 3/4 cup of the milk with the sugar and basil; bring to a simmer. Remove the milk from the heat and let stand for 15 minutes. Remove the basil and squeeze any milk back into the pan; discard the basil.

In a small bowl, whisk the remaining 1/4 cup of milk with the yolks and cornstarch until smooth. Slowly whisk the egg 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 the cream into a bowl. Press a piece of plastic wrap directly on the surface and refrigerate until chilled, 2 hours.

Preheat the oven to 375 degrees . Spray a 14-by-4 1/2-inch rectangular tart pan with a removable bottom with nonstick cooking spray. In the bowl of a standing mixer, beat the hard-boiled egg yolk with the butter and sugar at medium speed until smooth, about 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 side of the tart pan. Refrigerate the crust for 30 minutes, or until chilled.

Bake the crust for about 25 minutes, until golden. Transfer the crust to a rack and let stand until cooled, about 1 hour.

Increase the oven temperature to 450 degrees . 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 the apricots are tender and lightly browned. Let the apricots stand until completely cooled, about 30 minutes.

Unmold the crust and transfer it to a serving plate. Using a small offset spatula, spread the cream evenly in the crust. Arrange the apricots on the cream, cut sides down, and brush with the melted jam. Cut the tart crosswise into strips and serve at once.

To make a 9-inch round tart, press the dough into a 9-inch round fluted tart pan with a removable bottom. Use 10 apricots and 1/4 cup of sugar for sprinkling. Double the pastry cream. Bake for 40 minutes.