Make the crust: Pulse the flour, sugar and salt in a food processor. Add the shortening and 3 tablespoons butter and pulse into pebble-size pieces. Add the remaining butter and pulse until pea-size pieces form. Add the ice water; pulse a few times to combine. If the dough doesn’t hold together when you squeeze it, add more ice water, 1/2 tablespoon at a time. Turn out the dough onto a piece of plastic wrap and form into a disk; wrap tightly. Refrigerate until firm, at least 1 hour or overnight.
Roll out the dough on a lightly floured surface into a 12-inch round. Ease into a 9-inch pie plate. Tuck the overhanging dough under itself; crimp the edges with a fork. Chill until very firm, 20 minutes in the freezer or 1 hour in the refrigerator.
Preheat the oven to 350˚. Line the pie crust with parchment paper or foil and fill with pie weights or dried beans. Bake until the crust is light brown around the edges, about 25 minutes. Remove the parchment and weights and continue baking until the crust is light brown all over, 10 to 15 more minutes. Transfer to a rack to cool completely.
Meanwhile, make the filling: Break the cinnamon sticks in half and put in a medium saucepan over medium heat. Toast, stirring, about 5 minutes. Add the evaporated milk, ginger, nutmeg and cloves and cook, whisking, until the milk comes to a simmer, 4 to 5 minutes. Remove from the heat and let steep, 15 minutes. Discard the cinnamon sticks and add the pumpkin, sugar and salt; whisk until smooth. Whisk in the eggs until just combined.
Pour the filling into the cooled pie crust and bake until set and no longer wet and jiggly in the center, 50 to 60 minutes. (If the crust is getting too dark, cover loosely with foil.) Transfer the pie to a rack to cool completely.