Preheat the oven to 350°F (175°C).
Boil the sweet potatoes in a pot of water for 15 minutes or until they are fork-tender. Drain and mash them until smooth.
In a bowl, combine the mashed sweet potatoes, melted butter, granulated sugar, brown sugar, milk, eggs, cinnamon, nutmeg, and vanilla extract. Mix until well combined.
Pour the sweet potato mixture into the unbaked pie crust.
Bake in the preheated oven for 55 to 60 minutes, or until a knife inserted into the center comes out clean.
Allow the pie to cool on a wire rack before serving.