Preheat the oven to 400 degrees F (200 degrees C).
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 a 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.