This question is similar to another one already on SE but this one focuses on the task as a repair issue rather than as a remodeling one.
How can I cheaply and easily (I am a layperson) create a stable, opaque "replacement" for a broken glass pane in a(n inside) door?:

Apparently one of the doors in my flat is partially made of glass and someone (either the previous owner or the renovators-- the house management of course denies that it could be the renovators' fault) decided to simply paint over them in order to make them "unglass". The glass plate itself seems to be 2mm thick and the cutout in which the plate sits is about 2cm deep (measuring both sides).