Skin the yams and taro, then place each in separate pots of boiling water and cook until soft, 20 to 25 minutes.
When cooked, drain the water and combine the vegetables. Slowly add the milk to the mixture and blend until pureed. Add the sugar and continue to blend.
When thoroughly blended, place in freezer-safe container and freeze. Scrape the frozen mixture into cups to serve.
When blending hot liquids, first let cool for 5 minutes or so, then transfer to a blender, filling only halfway. Put the lid on, leaving one corner open; this will prevent the vacuum effect that creates heat explosions. Cover the lid with a kitchen towel to catch splatters and pulse until smooth.