Preheat the oven to 400 degrees F (200 degrees C). Press the pie crust into the bottom and up the sides of a 9-inch pie plate.
Bake the pie shell for 3 to 5 minutes, just to firm it up, then remove from the oven, and set aside. Place carrots in a saucepan with enough water to cover. Bring to a boil, and cook until tender, about 10 minutes. Drain water, and mash carrots until smooth using a food processor, or potato ricer.
In a medium bowl, mix together the carrot puree, sugar and eggs. Mix in the cinnamon and vanilla. Gradually stir in the milk. Pour the mixture into the partially baked pie shell.
Bake for 10 minutes in the preheated oven, then reduce heat to 350 degrees F (175 degrees C). Bake for an additional 40 to 45 minutes at the lower temperature, or until firm. Cool completely before serving.