Place whole sweet potato in pot and cover with water; bring to a boil. Boil until tender when pierced with a fork, 40 to 50 minutes.
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Preheat the oven to 350 degrees F (175 degrees C).

Remove sweet potato from the pot and run under cold water. Remove and discard skin.

Break sweet potato flesh apart and place in a bowl. Add butter and mix with an electric mixer until well combined.
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Add sugar, milk, eggs, nutmeg, cinnamon, and vanilla; beat on medium speed until mixture is smooth.
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Pour filling into unbaked pie crust.
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Bake in the preheated oven until a knife inserted in the center comes out clean, 55 to 60 minutes.

Remove from the oven and let cool before serving.
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