Clean the carrots and grate them, or chop them into a food processor or blender. Add the water and pulse until pureed. It is best to do this in two batches. Strain into a bowl using a fine strainer or cheese cloth. Squeeze the pulp to get all of the juice. You can strain a second time if you prefer less pulp in the juice.
Add the vanilla, cinnamon and nutmeg and stir well. Add the sweetened condensed milk to your taste. Start with 1/2 a can and add until it is sweet enough for you. Chill for a couple of hours and serve over ice.