Make the crust: Put the flour, sugar and salt in a food processor and pulse to combine. Add the butter and shortening; pulse just until the pieces are the size of large peas. While pulsing, gradually add the elderflower liqueur and 1/3 cup ice water until the dough comes together. Turn out the dough onto a lightly floured surface and knead gently until smooth. Divide the dough between 2 sheets of plastic wrap; use the plastic wrap to form each half into a disk. Wrap tightly and refrigerate until firm, at least 1 hour or overnight.
Meanwhile, make the filling: Toss the blueberries, sugar, flour, elderflower liqueur and salt in a large bowl. Let sit until juicy, 30 minutes. Position a rack with a baking sheet in the lower third of the oven and preheat to 425 degrees F.
Roll out 1 disk of dough into a 12-inch round on a lightly floured surface. Ease into a 9-inch deep-dish pie plate, pressing the dough into the bottom and up the side. Roll out the second disk of dough on a lightly floured surface.
Toss the berries again and pour the mixture into the prepared crust. Moisten the edge of the crust with cold water and lay the second dough round over the filling; gently press the top and bottom crusts together around the edge. Fold the overhanging dough under itself and crimp the edges. Cut a few slits in the top crust to let steam escape. Beat the egg with 1 teaspoon water. Brush the pie with the egg mixture and sprinkle lightly with sugar.
Put the pie on the hot baking sheet and bake 15 minutes. Reduce the oven temperature to 350 degrees F; bake until the crust is golden brown and the filling is bubbling, about 55 more minutes. (Cover the edge of the crust with foil if it is browning too quickly.) Transfer to a rack and let cool at least 4 hours before serving.