How does the Latin “SAPIAT” become “sache” [saʃ] in modern French?
I thought the reason may be palatalisation as [ʃ] seems to be a palatal sound, but I can't think of any other examples to prove this theory.
TL;DR: indeed, it is a kind of palatalisation, other examples include cavea > cage, rabia > rage, jam > déjà…
As usual in this case, I recommend the excellent Petit précis de phonétique historique by Noëlle Laborderie. In this case, sapiam's evolution is studied in chap. 16 3.2. I will make a short summary adapted to the case of sapiam. I will use IPA so, latinists, feel free to add your filthy transcriptions in Bourciez' alphabet if you feel such urge.
Here¹, palatalisation, means “a shift of the place of articulation toward the hard palate” which can result in a sound shift, for example the palatalisation of the [k] sound in Latin cabalus yields a [kʲ], then a [tʲ] and eventually a [tʃ] which lead to the [ʃ] in French cheval.
There are two main triggers of palatalisation in the study of the evolution of Latin into French, which are: reinforcement of an already palatal sound and assimilation with a contiguous palatal sound (usually the palatal approximant [j] or a front vowel).
In Laborderie's terminology, inherited from Georges Straka, a palatalisation is said to be true if it is due to both of these phenomenon and fake if not.
The accent of Classical Latin has affected most of the subsequent phonetic evolutions of Latin, without further details, let us admit that the accent in Classical Latin sapiam ['sapiam] was on ['sa], that the [i] was a so-called breve vowel and that all breve [i] and [e] evolved in [j] around the first century BCE, so with the unrelated loss of the final [m] it became [sapja]
The palatalisation of [j] is a fake one according to what we saw: since [j] is already palatal, what happened was only a reinforcement.
This kind of palatalisations happened around the III century CE and its result is in general an affricate, generally [d͡ʒ ]. But in this case, the [j] is right after a bilabial consonant, [p], which altered the palatalisation, due to the non-existent role of the tongue in its articulation. So instead of [d͡ʒ ], it yielded [ʈʲ], which then yielded [tʲʃ], [tʃ] (around the VII century CE) and finally [ʃ] around the XIII century CE.
You asked if there was other examples of this phenomenon. If found no other example of this precise occurrence [pj] > [tʃ]², but there are several other examples of fake palatalisations of [j], both in initial position, where the [j] was already present in Classical Latin such as iam > déjà or diurnu > jour and in intervocalic position, where it comes from a mutation of a [i] or [e] in hiatus, such as rabia > rage or cavea > cage.
¹ as everywhere else as far as I know.
² but then, I am by no mean a latinist