In a small bowl, cover the apricots with hot water and let stand for 10 minutes, until they are plump. Transfer the apricots to paper towels to drain.
In another small bowl, rub the tarragon leaves into the sugar until they are moist and aromatic. In a standing mixer fitted with the paddle, beat the butter with the tarragon sugar at low speed until creamy. Beat in the egg yolk until just combined, about 1 minute. Slowly drizzle in the olive oil and beat until smooth. Add the salt and flour and beat until just incorporated. Using a large spatula, fold in the apricots.
Turn the cookie dough out onto a work surface and knead until it just comes together. Divide the dough in half and press each half into a disk. Roll out each disk between 2 sheets of wax paper to about 1/4 inch thick. Cut out cookies with a 1/2 inch cookie cutter, leaving the cookies in the disk. Slide the wax paper-covered disks onto a baking sheet and freeze for at least 1 hour, until very firm.
Preheat the oven to 350 degrees and line 2 large baking sheets with parchment paper. Press cookies out from disks, reserving leftover dough to re-roll for more cookies. Place 1 inch apart on cookie sheet.
Bake cookies for about 20 minutes until they are lightly golden; shift the baking sheets from top to bottom and front to back halfway through. Let the cookies cool on the baking sheets for 3 minutes, then transfer them to a wire rack to cool completely.