Disclaimer: I did this as a shot in the dark. I don’t really know what I’m doing; I’m just throwing this out here on the off chance that someone can get some use out of it.
After reading Dawud’s post, above, I did some piddling around the on 'net today, and I finally found a Ministry of Education site that seems to contain some miscellaneous university job postings (i. e., not exclusively English-teaching positions):
https://tjn.moe.edu.tw/jobshow.php
But I don’t know whether that’s what Dawud was referring to, and if there are university job postings on the above-linked site, I don’t know how many there are.
I don’t know Chinese, so I had to use Google Translate, mostly its Chrome browser function.
Just as examples, I’ll just show the three that I noticed, the first two of which were advertised on the list part of the MOE job ad site as being in the school’s department of foreign languages, and the third of which (as it appears on the list part of the MOE job ad site) contains lingo which, even in translation, is above my pay grade. Anyway, here they are, bearing in mind that my Chinese ability is in the negative column, so it’s possible that these are actually three ads for hydraulic engineering, crop physiology, and music theory professors:
https://tjn.moe.edu.tw/jobshow-detail.php?num=5607
https://tjn.moe.edu.tw/jobshow-detail.php?num=5594
This is a link to a Google Translate version of the third one, which was a PDF document that I wound up plugging into the Google Translate site (apologies–I must have messed up the embedding job earlier, so the link didn’t work–I think I’ve fixed it now).
This is the original PDF document:
http://sites.cjcu.edu.tw/wSiteFile/File/B1600/171115151946106語文中心徵聘專案助理教授公告_20171115-20171215.pdf
On another subject, Wikipedia says this is “a list of universities, colleges, junior colleges and institutes in Taiwan”:
The Ministry of Education appears to have some downloadable lists of colleges and junior colleges in Excel spreadsheet format:
http://english.moe.gov.tw/ct.asp?xItem=552&CtNode=415&mp=1
I hope this helps, or at least does no harm.