I am not an expert or a native speaker, but here is my understanding.

The phonemic contrast in French between /e/ (as in né) and /ɛ/ (as in lait, muet, muette) is fairly variable and unclear in non-final syllables.

In fact, even « é » in a non-final syllable is not always pronounced as /e/; some speakers pronounce « é » as /ɛ/ in some contexts. (In reformed spelling, some but not all words have spelling variants with « è » to reflect this, like évènement in place of the traditional spelling événement).

Two factors that can influence the pronunciation of « ai », « é » and « ê » in non-final syllables are syllable structure and vowel harmony.

In épaissir and laisser, the vowel in the syllable following « ai » is close or mid-close. This influences some speakers to pronounce the « ai » as mid-close [e] rather than as mid-open [ɛ]. Similarly, « ê » can be pronounced [e] in certain words, like bêtise, even though « ê » in the final syllable of a word usually corresponds only to [ɛ].

Likewise, there are speakers who pronounce aimer as [eme], but aimable as [ɛmabl], because in aimer the vowel in the following syllable is mid-close [e] but in aimable the vowel in the following syllable is open [a]. But as far as I know, all speakers pronounce aime as [ɛm(ə)]; likewise, all speakers would I think use [ɛ] in the last syllable of épaisse. (With épais, the situation gets a bit more complicated: /ɛ/ in the last syllable is definitely standard, but because the vowel comes at the end of the word in an “open” syllable (with no consonant sound following it), speakers in a few regions may tend to pronounce the vowel in a way that sounds more like [e].)

The user Eau qui dort left a comment beneath another one of my posts that talks about this: Pronunciation of the « é » in « médecin »

I don’t think the use of [e] vs. [ɛ] in contexts like épaissir and laisser is a big deal. There are regional variations, but as far as I know it is not a major marker of a regional accent.

Interesting question, epaissir et laisser, it all depends and both are correct
i would say épéssir and lésser and also épèssie et lèsser or émmer or émmer.

when you speak you tend not to focus to much on certain sounds, it could be a sound between é and é, it is quite subtle

As a french speaker, my knee-jerk reaction would have been to say /ɛ/ in all cases. That is how I hear the words in my head while reading them, and how I learned their pronunciation in my youth.

However, I tried pronouncing them out loud, and it seems that I would say “épéssir” and “lésser” when in the context of a full sentence. On their own, I’d pronounce them “lèsser” and “épèssir”.

In any case, what I learned when I was younger is that -ai was always pronounced /ɛ/.