Whirl flour, powdered sugar, and salt in food processor with the blade attachment. Add slices from one stick of butter and whirl until no chunks remain. Add remaining slices and whirl briefly to chop them up slightly. Turn food processor on again and quickly begin drizzling ice water through the feed tube, stopping before the dough comes together. Pinch some crumbs together between your fingers; if they don’t stick together, add a bit more water and whirl briefly.
Turn the mixture out onto a large piece of plastic wrap. Use the plastic wrap to help you bring the dough together; fold the edges of the dough toward the center and press down until it holds together, taking care not to overwork the dough. Divide it in half and shape into two flat disks wrapped tightly in plastic wrap. Refrigerate dough for 30 minutes.
While the dough chills, combine apricots, sugar, nutmeg, lemon juice, salt, and cornstarch. Let stand for 15 minutes, stirring occasionally.
Roll out one of the dough rounds into a 13- to 14-inch circle. Transfer carefully to a 10-inch pie dish. Roll out remaining round into a 13- to 14-inch circle. Fill bottom crust with apricot mixture. Brush the overhanging edge with cold water, cover with the top crust, and seal gently by pressing down on the edge with the bottom of a fork. Trim overhanging dough with a knife and flute the edge with your fingers, if desired. Cut steam vents in the top.
Whisk egg and milk or cream together in a small bowl. Lightly brush the top of the pie with the egg mixture. Sprinkle evenly with 2 teaspoons of sugar.
Place pie on a baking sheet to catch any overflowing juices. Bake in the bottom third of an oven preheated to 425° F for 30 minutes. Reduce heat to 350° F and continue baking for 30-45 minutes more, until juices bubble thickly through the vents and the crust is pale golden brown (if the top browns too quickly, slip a baking sheet on the top rack of the oven to diffuse the heat). Allow to cool for several hours before serving.