The crust. To make the dough, place the all-purpose flour, salt, and cold butter in a food processor and pulse until it resembles coarse breadcrumbs. Add cold water, and process until moist clumps form. If the dough doesn’t clump, add a little ice water by teaspoonfuls and pulse to incorporate. It’s important not to add too much water to the dough, or the dough will be sticky and difficult to roll out.
Gather the dough into a ball and knead it several times on a lightly floured surface. Form the dough into an even disk, cover with plastic wrap and refrigerate for about 30 minutes.
The filling. In a large bowl, whisk together the eggs, granulated sugar and brown sugar until well combined. Stir in the melted butter, vanilla and heavy cream. Add the mashed purple sweet potatoes and whisk until incorporated and smooth. Stir in the cinnamon, ginger, allspice and nutmeg.
Assemble. Flour the working surface. Roll the dough to a circle 1/4-inch thick. Keep rotating the dough while rolling to prevent it from sticking to the surface.
To transfer the dough, carefully roll the dough around the rolling pin, then transfer it to the pie dish and unroll. Press the dough into the edges and corners, being careful not to tear it. Trim the dough about 1 inch larger than the dish. Fold the edge under and crimp the edge with your fingers. Chill in the refrigerator for about 30 minutes.
Preheat the oven to 375°F (190°C). Line the chilled pie crust with parchment paper and fill it with pie weights or dried beans. Bake until the edges of the crust are starting to brown, about 15 minutes.
Take the crust out of the oven and carefully remove the weights by lifting up the parchment paper. Brush the crust with the egg wash and prick the bottom of the crust all over with a fork. Return the pie crust to the oven. Bake until the bottom crust is just beginning to brown, about 7 minutes. Remove from the oven. Cool slightly.
Pour the prepared filling into the partially baked pie crust and bake for 40-45 minutes, until the filling is set at the edges but still slightly jiggly in the center. If the crust is browning too quickly, tent a piece of aluminum foil over the pie to prevent the crust from burning.
Remove from the oven and cool completely.