Preheat your oven to 400°F (200°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.
Remove it from the oven and set it aside.
In a saucepan, place the chopped carrots with enough water to cover.
Bring to a boil and cook until the carrots are tender, about 10 minutes.
Drain the water and mash the carrots until smooth using a food processor or potato ricer.
In a medium bowl, combine the carrot puree, sugar, and eggs.
Add the cinnamon and vanilla, mixing well.
Gradually stir in the milk.
Pour the mixture into the partially baked pie shell.
Bake for 10 minutes in the preheated oven, then reduce the heat to 350°F (175°C).
Continue baking for an additional 40 to 45 minutes at the lower temperature or until the pie is firm.
Allow the pie to cool completely before serving.
