Taiwan is a place where it's in a state of constant flux. Shaved ice is traditional and old in these parts. But in Taipei these days, traditional shaved ice exists mostly as a nostalgic factor... sort of how we have '50s Diners. Most places serve Western style ice cream instead. There was a time, maybe in the 70s and early 80s,when people's taste wanted a Western ice cream but milk was still an expensive item, when the ingenious Taiwanese invented ice sheets as a compromise. It's basically water with a bit of milk thrown in to it give a slightly milky taste. We had the hardest time finding this confection and got lots of raised eyebrows for asking for it. Apparently we didn't get the memo that it was now passe and gauche.
We finally found a place near Shihlin Market that still sells it and it was divine. We got the tea flavoured with red beans. A bit of condensed milk tops it all.
The plain, undressed shaved ice sheets.