Can gemstones be cut into sheets?
sliced (like glass window)


Window glass is not sliced into sheets. It is made as sheets from the beginning.

Since you're looking for a naturally occurring mineral that can be made into a glass-like material, look no further than muscovite.



Because your planet has nice and big crystals, it most likely come from pegmatites, which is a type of rock that has large crystals and many gemstones we know today come from pegmatites (emerald, tourmaline, topaz, kunzite, etc). Pegmatites also have huge crystals of muscovite as well, in some rare occasions several metres wide. Muscovite can be easily split into sheets that are flexible and mostly transparent. Not the best glass material (because it is not glass after all), but almost as good. Quoting from the Wikipedia page:


  The name muscovite comes from Muscovy-glass, a name given to the mineral in Elizabethan England due to its use in medieval Russia as a cheaper alternative to glass in windows. This usage became widely known in England during the sixteenth century with its first mention appearing in letters by George Turberville, the secretary of England's ambassador to the Russian tsar Ivan the Terrible, in 1568. 


Here's a good video showing how this is working.

So the inhabitants of your planet can mine the other crystals in the pegmatite for currency or whatever, and use the muscovite as building material.