I have successfully dealt directly by phone with the CX office in Taipei for a DONE4 and arranged for the ticket to be waiting for me at the airport ticket desk for pickup on a same plane turnround.
https://www.cathaypacific.com/cpa/en...countryCode=TW

(Actually, strictly, it was my secretary and I suspect she was talking to them in Chinese, so I'm not sure how well they function in English...)