Bake the sweet potatoes - Arrange a rack in the middle of the oven and heat the oven to 375°F. Place the sweet potatoes directly on the oven rack (place a piece of aluminum foil onto the rack below if you're worried about drips). Bake the sweet potatoes until tender, about 1 hour. While the potatoes bake, prepare the crust for baking.
Prepare the pie crust - Roll out the pie crust into an 11-inch round. Transfer it to a deep, 10-inch round tart pan and use it to line the bottom and sides of the pan, pressing the dough into the sides as needed. Refrigerate while the sweet potatoes bake and while preparing the filling, at least 30 minutes.
Cool the sweet potatoes and scoop out their flesh - Cool the potatoes for about 30 minutes, or until you can comfortably handle them. Cut the sweet potatoes in half lengthwise and scoop the flesh out into a food processor fitted with the blade attachment. (Alternatively, store the sweet potato flesh in the refrigerator for up to 1 week before proceeding.)
Purée the sweet potatoes - Process the sweet potatoes until very smooth.
Make the filling. Add the brown sugar, vanilla, cinnamon, ginger, nutmeg, salt, eggs, and evaporated milk to the food processor and pulse to combine.
Fill the crust - Place the tart pan on a parchment-lined baking sheet. Pour the filling into the crust. The filling should come almost to the top of the crust, but be carefully not to overfill. You may have a 1/4 to 1/2 cup of extra filling, which you can bake alongside the pie in an oven-safe ramekin, if desired.
Bake until the center is just set - Bake until the top appears dry (it may puff and crack slightly around the edges) and the center jiggles, but doesn't wiggle, when nudged, about 45 minutes.
Cool - Cool the pie completely before serving — it will continue to set as it cools.