Step 1Heat the oven to 350 degrees. Sprinkle a work surface with a thin layer of flour. Slightly flatten the ball of dough and shape into a square. With a lightly floured rolling pin, roll out the dough to slightly less than one-fourth-inch thick and about 12 inches square. Roll the dough around the rolling pin and carefully lift it into the tart pan. Lift the edge of the dough with one hand; use the other to gently press the dough into the bottom and against the sides of the pan. Trim the edges; you will have dough left over. Chill the tart shell in the refrigerator for 30 minutes.
Step 2Blanch the apricots, a few at a time, in boiling water for 30 to 40 seconds to loosen the skin. Immediately drain the apricots and plunge in a bowl of ice water to cool. Peel the skins from the apricots, and halve lengthwise, removing the pits.
Step 3In a large saute pan or shallow stockpot, measure enough water to come 2 to 3 inches up the sides of the pan. Stir in the same amount of sugar as water to make a simple syrup. For example, if you used 6 cups water, then stir in 6 cups sugar. Over medium-low heat, bring the mixture to barely a simmer. Poach the apricot halves for 1 to 2 minutes (time will vary depending on the ripeness of the apricots) until fork-tender. Drain and cool on a lipped baking sheet.
Step 4In a mixing bowl, cream the softened butter with a wooden spoon. Add the sugar, ground almonds and flour. Mix until all ingredients are thoroughly blended.
Step 5In a small bowl, beat one of the eggs and set aside. In a separate bowl, beat the remaining egg. Add one beaten egg to the butter-sugar-almond-flour mixture, beating all the time. Then add half of the other beaten egg (about 2 tablespoons; you can discard the rest of the egg) to the mixture. Add the pistachio paste, mixing well.
Step 6Remove the tart shell from the refrigerator and place on a baking sheet. Pour the pistachio mixture into the pastry shell. Gently pat the apricots with paper towels to remove excess moisture. Arrange the apricots, cut side up and overlapping, and cover with the rest of the filling. Bake 50 minutes to 1 hour, until the filling is set and beginning to color (it will be golden in places). Cool slightly and sprinkle with chopped pistachios before serving.