Several years ago, I had the best shaved ice EVER in the Shiling night market in Taipei. The toppings and sugar liquids were the same, but instead of shaving frozen water to create the ice, they used a frozen block of sweetened condensed milk(probably diluted with some water).
Instead of the snow like consistency of regular shaved ice, it shaves off in very very thin layers and each layer settles on top of the other layer, looking almost like whipped cream but without the density and thicknenss of whipped cream. The taste is heavenly -- without the crunchiness of regular shaved ice but still having the melt-in-your-mouth feeling of shaved ice. The standard toppings(green beans, ai yu jelly, sugar water, sweetened condensed milk, etc.) were added to this base of "shaved milk" but I would have happily eaten the ice plain.
The Chinese name for this milk ice is Xue Hua Bing(literally Snow Flower Ice) while the Chinese for regular shaved ice is "Cua Bing."
I searched all over the Taiwanese spots in Southern California but was never able to find this. If anyone knows anything about this, please share.