The example quasi-first-class fits the following pattern that The Chicago Manual of Style, 16th edition (2010) addresses at 6.80:
A single word or prefix should be joined to a hyphenated compound by another hyphen rather than an en dash; if the result is awkward, reword.
a two-thirds-full cup (or, better, a cup that is two-thirds full)
But notice that, in the first instance, the prefix non- is added to a hyphenated compound only because English-speaking appears as a compound adjective before the noun peoples. If the construction had been "peoples who are English speaking," there would have been no hyphen between English and speaking—and in that case, the negative of "English speaking" would have needed to be non– with an en dash:
peoples who are non–English speaking
to indicate that the prefix non- applied to the two-word phrase English speaking and not merely to the single word English. In other words, the subject here is peoples who do not speak English, not peoples who speak non-English.
Analogously, in the poster's original example of quasi-first-class, Chicago seems to endorse the two-hyphen approach when the compound adjective precedes the noun:
a quasi-first-class operation
but a single en dash when the modifier follows the noun:
an operation that is quasi–first class
The accepted answer to this question does not address the relevance of where the noun falls in relation to the compound (or multiple-word) modifier, but in Chicago's treatment of the issue, the position of the modifying phrase relative to the noun is extremely important.
Other style guides may take different positions on this issue—and many appear to take no distinct position at all—but Chicago's guideline at least has the advantage of supporting a consistent approach to compounds of the sort that the poster asks about.