From my viewpoint,
1) "a quorum of at least half of the 20 directors" is the subject, means "at least half of the 20 directors should be present to the meeting and those are the persons who will vote", that is the meaning of the word "quorum" .
2) "must pass the resolution " is the verb and its object, that is clear.
3) "by at least a two-thirds majority." is describing how the resolution could be passed "2/3 majority".