In a medium saucepan, combine the sweet potatoes, cinnamon stick pieces, cloves, star anise, and ginger slices with enough water to just cover the contents. Bring to a boil over high heat. Reduce the heat and simmer uncovered until the sweet potatoes are very tender when pierced with a fork or skewer, about 10 minutes. Drain the potatoes, reserving the boiling liquid. Return the potatoes to the pot over low heat and toss to dry them a bit. Discard the cinnamon, cloves, and star anise. Force the warm potatoes through a ricer, a food mill, or a sieve. Boil the liquid if needed, until reduced to 1/4 cup. Let the sweet potato mash and the liquid cool.

Position an oven rack in the lower half of the oven; heat the oven to 350 degrees F. In a large bowl, whisk together the pumpkin and sweet potato puree. Whisk in the egg, egg yolk, melted butter, and reserved spiced liquid. In a separate bowl, stir together the granulated and brown sugars with a wire whisk until any large lumps of brown sugar are gone. Sift the flour and salt over the sugars; stir to blend. Add the sugar-flour mixture to the pumpkin and stir well until no pockets of sugar are visible. Blend in the coconut milk.

Scrape the filling into the chilled pie shell; smooth the top. Bake for 1-3/4 to 2 hours, turning the pie several times so it bakes evenly. The point of a thin-bladed knife should come out clean when inserted into the center of the filling, and the edges of the surface will be unevenly cracked. If the edges of the pastry darken too much before the filling is cooked, cover them with a pie shield or strips of aluminum foil. Transfer the pie to a wire rack and let cool completely before serving with mounds of the lightly sweetened whipped cream.

Make Ahead: The sweet potatoes and spiced liquid can be prepared up to three days ahead and refrigerated. Bring each to room temperature before proceeding with the recipe. Each of these elements can also be frozen for up to three months; thaw overnight in the refrigerator before bringing to room temperature.

Be sure the pastry is still well chilled when you put the pie in the oven. This helps to ensure that the crust won't darken too much during the long baking time.

Combine the flour and salt in a large mixing bowl or a food processor. Add the shortening and cut it in with a pastry blender or two knives (or pulse the processor) until the largest pieces of shortening are about the size of fat peas. Transfer to a large mixing bowl if using the food processor. In a small bowl, beat together the egg, 2 tablespoons of the ice water, and the vinegar; add this to the flour mixture. Work the liquid evenly through the dough with the tips of your fingers until it can be collected in a rough ball. If the dough is too dry to come together, gradually sprinkle a few drops of the remaining water over it and continue to work gently until it comes together.

Cut the dough in half and shape each into a ball. Flatten each ball into a disk and wrap tightly in plastic. Chill for at least 4 hours or up to three days. (Freeze one if you like, to save for another purpose.)

About 2 hours before baking, remove one of the dough disks from the refrigerator so it can warm up just enough to roll without cracking. Unwrap the dough and roll it between two sheets of waxed paper into a round about 12-inches in diameter. Peel off the top sheet of waxed paper. Invert a 9-inch glass pie plate and center it on the dough. Slide one hand under the bottom sheet of waxed paper, position your other hand flat on the pie plate, and quickly flip the plate and dough over. Peel off the waxed paper and gently press the dough down into the plate. With a paring knife, trim the dough to within 3/4-inch of the rim of the plate. Fold the dough under and crimp the edges. Use the trimmings to patch any cracks or bare spots. Wrap the shell loosely with plastic wrap and chill for 2 hours or up to 24 hours.