Preheat oven to 325 degrees F (165 degrees C). Grease a 1 quart casserole dish.
Place carrots and enough water to cover them in a small saucepan. Bring to a boil, and cook until carrots are very tender. Drain. Using a blender or food processor, puree carrots to a smooth consistency.
In a medium bowl, mix together the carrots and butter. Beat in the eggs, sugar, milk, salt, cinnamon, nutmeg, and vanilla. Spread mixture into the prepared casserole dish.
Bake for 25 to 30 minutes in the preheated oven, or until set.