There is an interesting early Christian document called the Διδαχή, translated into English as "The Teaching." The word seems to be classical, not just Koine. Is this some kind of more general "that which" or "activity involving" construction? If so, is it constructed from a verb like διδάσκω, or is it some kind of noun construction like English "youtubing?"
διδαχή is indeed built on διδάσκω, though without the inchoative infix -σκ-: the root is διδαχ-, as can be seen in the aorist ἐδίδαξα, and -σκ- is one of those infixes that only show up in the present. The reduplication, in this case, indicates a causative (the original verb is the unattested-in-the-present *δάω 'to learn', though it does show up in the aorist as ἐδάην), and for semantic reasons that is preserved (and, indeed, piled onto in the perfect: δεδίδαχα).
Greek has a lot of suffixes that form nouns out of verbs, but the simplest one is just the first-declension ending ‑η. This was a mechanism that was already available in Proto-Indo-European (when the ending was still *‑eh₂) where it could be added to the o-grade of verb roots (as in the Greek descendants τρέφω 'to nourish' → τροφή 'nourishment'; διαφέρω 'to differ' → διαφορά 'difference' (ᾱ instead of η after ι, ε, or ρ in Attic); λείβω 'to pour' → λοιβή 'libation'), the e-grade (στέγω 'to cover' → στέγη 'roof, shelter' (note the accent)), or the zero-grade (ἄρχω 'to begin; to rule' → ἀρχή 'beginning; rule'; φεύγω 'to flee' → φυγή 'flight'), but the construction remained productive well into historical times. διδαχή is formally built on a zero-grade (root *dens-, zero-grade *dn̥s-), but really it's post-PIE, when those distinctions no longer make much synchronic sense.
Though this is often casually described as creating verbal nouns that's misleading: the meaning, for a verb X, isn't the act of doing X so much as an instance of Xing, as you can see. διδαχή is a teaching, not the act of teaching.