Place taro roots in a small pot; cover with water. Bring to a boil. Cook taro until tender, about 12 minutes. Drain.
Combine taro roots with 1/2 cup ice in a bowl. Let cool.
Combine cooled taro roots, remaining 2 cups ice, milk, 3/4 cup water, sugar, and 1/2 teaspoon honey in a blender; blend until smooth.
Bring 2 cups water to a boil in a saucepan. Add tapioca pearls; stir gently until they start floating to the surface. Cook for 12 to 15 minutes. Remove from heat, cover, and let stand until tender, 12 to 15 minutes more. Drain. Stir in remaining 1 teaspoon honey.
Divide blended mixture between 2 glasses. Divide tapioca pearls between the glasses.