American in Beijing/overseas Taiwan passport/job offer in Taipei. What now?

I know this topic has been partly covered in other threads, but I can’t seem to find the solution to my specific case.

I am a U.S. passport holder in Beijing (female, 32 years old). I have a Chinese Z visa (work and residence), and I also have an overseas Taiwan passport (no hukou, no shenfenzheng). I have never lived in Taiwan, nor have I ever entered on my Taiwan passport before (it’s new). However, I have just been offered a job in Taipei and would like to move there and start the process for residence. Also, for reference, my mother is Taiwanese, with hukou and shenfenzheng.

Does anyone what I should do to get the ball rolling? It seems that I should get an entry permit, enter Taiwan, then apply for a TARC (is this right?). If this is the case, does anyone know if I can get an entry permit in Beijing, or will I have to go somewhere else? Also, are there any other obstacles, given my situation, to starting my residence process?

My ultimate aim is to get both my hukou and shenfenzheng sorted so that I can properly establish myself in Taiwan.