So actually there have been a few cases where extensions have been applied for and the application process has gone beyond the expiry date of the original Gold Card though due to how an extension works, when approved the start date of the new Gold Card is “backdated”, therefore valid residency is continuously maintained for APRC.
Extension Application: Refers to submitting an extension application within four months before the Gold Card expires. The start date will immediately follow the expiration date of the previous Gold Card.
What is the difference between re-application and extension? | Taiwan Gold Card
Even though this is possible I usually recommend people to apply early as between the original Gold Card expiring and the application being approved is somewhat of a grey area if the ID was checked or had to leave the country and also now with telecoms checking for a valid residency. Additionally if the application is rejected then essentially in a possible overstay situation unless a 6 month extension was given.
All cases I’ve heard of so far though that ended up in that situation eventually got approved.