Learn English – the comparative and superlative form of “super”

That "rule" is a very crude approximation of what actually happens.

For most 2-syllable adjectives, either form (more/most or -er/est) is at least "credible" to most if not all speakers, but for any specific word the relative frequency of one may be slightly or significantly greater.

You can add extra "general principles". For instance, two-syllable adjectives ending with –y and –ow, readily take the –er/–est endings, but those with –le and –er characteristically don't for some speakers.

Finally, there are even a few acceptable 3-syllable forms - unlikeliest and unhappier, for example. The un- prefix seems to favour "special dispensation to buck the basic rule", but with apparently 1330 instances in print for almightiest I think we have to accept that one as "credible". With no discernable "extra principle" - it's just a "one-off" that doesn't seem to conform to any rule or exempting principle.

Bear in mind that for any given pair of native speakers it's quite possible they will disagree on the acceptability of certain -er/-est forms.

In the specific case of handsomer,more handsome, as you'll see from that link, usage has changed dramatically over the past century. The latter, more "generic" form is now actually the most common, but C19 usage was dominated by handsomer. What this shows is that people are gradually moving towards implementing the simple rules more consistently, but it's a slow process. Nevertheless, on average we're becoming more likely to favour more/most, and unlikelier to use forms like that

As this link shows, even though I'm presumably unassailable in having used more common above, a substantial minority would have been perfectly happy with commoner not so long ago.

I don't think the average "learner" really needs to know that some people still find handsomer acceptable (most don't, and you'd never be criticised for saying more handsome, so just do that anyway). I suggest using the more/most forms for all 2-syllable adjectives except where the second syllable ends in /i/ (easy, happy, silly), or the second vowel is a neutral schwa (clever, humble). And I'd call quiet a single-syllable "triphthong", which for me explains why quieter/quietest are okay. But if in doubt, just use more/most.

Both of your examples

This is done easily.

This is done more easily than I thought.

are grammatically correct.

However, I would personally consider both sentences to be slightly awkward or incomplete, since you could express the same ideas using adjectives. If you had no more details to add, then more natural expressions might be

This is easy.

This task is easy.

This was easier than I thought.

This job was easier than I thought.

The original examples would also be fine if there were more details to complete the sentence, so that there is a reason to use the adverbial form rather than the simpler adjectives that I proposed above.

Calculating this antiderivative is easily done using integration by parts.

Calculating this antiderivative is more easily done using a trigonometric substitution.