Learn English – Proper verb for power outage

In short, your only option seems to be exegese (or possibly exegeze, but probably not), if you must have a verb that would be acceptable to the OED.

But it should be noted that, for what is probably the majority of words on -sis, no verb is ever used; perhaps recasting your sentence would be a better idea, e.g. she interpreted the passage. Using the Greek infinitive exegeisthai is a bad idea that will send classicists off screaming.

The only traditional way of forming a verb based on a Greek word on -sis that I am aware of is -se:

Parenthesis => parenthese

Diagnosis => diagnose

Paralysis => paralyse

Basis/-se => base

Metamorphosis => metamorphose

Occasionally, modern forms on -sise are created, as in those based on -thesis (hypothesise, synthesise); I can't think of any other example. Those few Greek words on -sis whose noun-equivalents already end on -se in English also use -se for the verb, as in phrase and base.

For forms on -ise and -yse, you may encounter -ize and -yze in America, respectively, and possibly sometimes elsewhere too, so exegeze may be considered a valid alternative depending on your local tradition. However, this strictly has nothing to do with the formation of an English word based on Greek, but rather on regional variations within English. Lastly, I do not believe Americans normally do this with words on -ese/-eze (they stick with -ese: only after -i- and -y- is this z commonly used), but that's your call.

Your examples ostracize and baptize are different, because they have somehow retained or regained the z that was originally there in Greek (ostrakizein, baptizein), as opposed to in exegeisthai and most other verbs. The same applies to apologize/apologizein. This confusion is the reason (most?) British publishers and style books use -ise/-yse for verbs based on Greek nouns on -sis (Oxford and Cambridge alike, I think), like analyse; but they do often do use -ize for other Greek verbs, like baptize, and for words taken from non-Greek stems, like immunize, realize, colonize (either Oxford or Cambridge—I forgot). They usually do not, however, use -yze where Greek had no -uzein (so practically never). I know, it is a bit tiresome having to remember the exact origin of such words—I am usually too lazy to do so, I must confess.

The -i- in *exegise/exegize is not really defensible, because that is normally not done with words derived from -esis, but only with those derived from -isis. The -ize in energize comes from Greek -(e)izein, as energy comes from energ(e)ia; the -i- in apologize from Greek -izein as in apology, from apologia. The -y in English represents -(e)ia in most Greek (and Latin) words, and hence nouns on -y in English have verbs on -ise/ize.

Aside from the rather intricate rules for what to do with the understood subjects of the second verb in these complex sentences, mentioned by Lawler in his comment, there is also a 3 way choice of complement type. What follows "consider" in the illustration you gave is taken to be a sentence, schematically [Lucas consider [Lucas bring back Hamill]], where the smaller sentence is called a sentential complement to the main verb "consider".

The 3 choices for complement type in English are that-clause, for-to, poss-ing. (I hope I didn't forget any.) (1) [Lucas consider [that Lucas will bring back Hamill]], (2) [Lucas consider [for Lucas to bring back Hamill]], (3) [Lucas consider [Lucas's bringing back Hamill]]. So part of what you're asking, I take it, is what principles determine for a given main verb ("consider" in the example) which of the three complement types will be possible, and what nuances of meaning will be expressed by the choice.

I'm sorry to have to report that the answer is unknown, and it may well be that there is no answer. English speakers may have to learn ad hoc for each complement-taking verb which complements are possible and how to interpret them. So for an English language learner, in this particular regard, grammatical study is probably not useful, and lots of experience with interpreting and speaking or writing English is all that can help.

The name of the relevant field of study, following Noam Chomsky in Aspects of the Theory of Syntax, is "the problem of verb subcategorization". It was tackled by George Lakoff in his dissertation Irregularity in Syntax, and there is a very good more recent discussion in McCawley's The Syntactic Phenomena of English.