For the crust: Put the flour, sugar and salt in a food processor fitted with a metal blade and pulse to combine. Add about a third of the chilled butter cubes and process until the butter is absorbed. Add the remaining butter and pulse until the butter is the size of peas, 5 or 6 pulses. Add the vinegar. Then, while pulsing quickly a few times, add the ice water through the tube. Squeeze some dough between your fingers; it should just hold its shape without feeling wet. If it is crumbly, add more ice water a tablespoon at a time. Pulse quickly; do not overwork the dough or the crust will be tough. Wrap the dough loosely in a large piece of plastic wrap, then firmly press and flatten the dough into a thin round. Refrigerate for 1 hour.
Lightly dust a work surface with flour. Then roll the dough on the floured surface into an 11- to 12-inch round, about 1/8 inch thick. Center the dough in a 9-inch pie plate, with an even overhang all around. Fold the edges and flute or crimp. Chill for at least 30 minutes.
Preheat the oven to 375 degrees F.
Press a piece of foil over the chilled crust, then fill with pie weights, raw beans or rice. Bake to set the crust, about 20 minutes. Remove the foil and weights and bake until the crust is light brown, 5 to 10 minutes more. Let cool slightly before adding the filling.
For the filling: Sprinkle the chives evenly in the crust. Combine the Parmesan, sopressata, and olives in a medium bowl and sprinkle in an even layer over the chives.
For the custard: Whisk together the heavy cream, eggs, 1/2 teaspoon salt and 1/4 teaspoon pepper in a bowl. Stir in the basil.
Pour the custard over the filling. Lower the oven temperature to 350 degrees F and bake until set, 35 to 50 minutes. Let cool at least 30 minutes before serving warm or at room temperature.