Make the crust: Pulse the flour, granulated sugar and fine salt in a food processor until combined. Add the shortening and pulse until incorporated. Add the butter and pulse until the mixture looks like coarse meal with pea-size bits of butter. Beat the egg and 1 tablespoon ice water in a small bowl; add to the food processor and pulse once or twice to moisten the flour. Turn out onto a piece of plastic wrap; use the plastic to help form the dough into a disk. Wrap tightly and refrigerate until firm, at least 1 hour or overnight.
Roll out the dough into a 12-inch round on a lightly floured surface. Ease into a 9-inch pie plate, then fold the overhanging dough under itself and crimp the edge with your fingers. Refrigerate at least 30 minutes.
Position a rack in the lower third of the oven and preheat to 350 degrees F. Make the filling: Spread the walnuts on a baking sheet and bake until toasted, about 8 minutes; transfer to a bowl and set aside. Return the empty baking sheet to the lower oven rack. Heat the butter in a small skillet over medium-high heat until foamy and golden brown, about 5 minutes; remove from the heat. Whisk the eggs, corn syrup, brown sugar, granulated sugar, kosher salt, liqueur and vanilla in a medium bowl. Gradually whisk in the browned butter until smooth.
Scatter half each of the walnuts and chocolate in the prepared crust. Pour in the filling, then top with the remaining walnuts and chocolate. Transfer the pie to the hot baking sheetin the oven. Bake until the crust is golden and the nuts have browned, about 30 minutes. Cover the pie loosely with foil; continue baking until the filling is set and the center is slightly puffed, 25 to 30 more minutes. Transfer to a rack to cool completely.