Place the sweet potato in a pot and cover with water. Bring to a boil and cook until tender when pierced with a fork, about 40 to 50 minutes.
Preheat the oven to 350°F (175°C). Remove the sweet potato from the pot, run it under cold water, discard the skin, and break the flesh apart into a bowl. Add butter and mix with an electric mixer until well combined.
Add sugar, milk, egg, nutmeg, cinnamon, and vanilla. Beat on medium speed until the mixture is smooth.
Pour the filling into the unbaked pie crust.
Bake in the preheated oven until a knife inserted in the center comes out clean, about 55 to 60 minutes.
Remove from the oven and let cool before serving.
Notes
For a twist, top the pie with whipped cream or a sprinkle of cinnamon before serving.