I think that you are confusing a number of related things here, particularly the issues of "resident visa" and "work permit". There are several qualifying categories whereby you may apply for a "resident visa", and one of the most popular is due to the fact that you have a "work permit" to work at a particular company or organization in Taiwan. According to the Employment Services Act (ESA), the employer must apply for your work permit, and submit all relevant documentation to the appropriate central government authority. In the case of a university, the appropriate central government authority would most likely be the Ministry of Education. If your qualifications are considered adequate, and the type of job you wish to undertake is one of the allowable categories as specified in ESA Article 43, then (hopefully) your work permit will be granted. At that point you can take your work permit approval letter to the Bureau of Consular Affairs of the Ministry of Foreign Affairs in Taiwan and apply for a resident visa based on that approved employment status. It is often heard that at that point you have to leave the ROC area, and then come back in on the new resident visa, but there may be some exceptions to that rule.
In general, if you want to get a work permit to work in Taiwan, and then to use that status to apply for a resident visa, your first step is to find an employer who is willing to hire you, and then have him/her go through the appropriate steps to apply for your work permit. It is not a "quick" process, and in fact may take a month or two in some cases.
Resident visas are based on one of the following categories: approved investment, approved employment, marriage to ROC citizen, approved missionary activities, or full time study in a recognized ROC educational institution. Basically, there are only five categories of resident visas for ordinary foreigners.