While give would be quite acceptable, between the two words, I would choose offer. Not only is it a good solid businesslike word, it is also slightly less demanding, and makes you seem less importunate and more polite.
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.
Best Answer
Onboarding
Wikipedia defines this as
"She has been tremendously helpful onboarding the company's new employees."
In my experience this term is very common in the HR field and is often used during the new hire process. My view of onboarding includes any type of activity or process that is aimed at preparing a new employee for his/her position. It can include training, getting certifications, briefing on policies and so on. I assume the process varies widely depending on the position and field of work.