If you're truly serious about working for an airline that is requesting certification if your hours, I strongly encourage you to have your chief pilot certify your logbook. It's not at all an unusual request, and the chief pilot will know exactly what to do.
If you cannot meet with your chief pilot, contact Asiana and ask what they require in lieu of flight hour certification. Some companies will accept a written letter (again signed by your chief pilot) verifying your flight hours.
My point is that Asiana is looking for certified flight hours from your previous or current employer. They want a signature certifying your flight time. Simply mailing, faxing, or scanning your flight hours in from your previous or current employer, most likely, will not meet the standards they are demanding.
Of course, this is all just advice. Others will have their own opinions and advice. My advice is coming from 4 invitations to interview at 4 foreign carriers. 2012-2016 time frame. 2 I never went to interview after deciding I wouldn't want to work there. 1 I interviewed and was ultimately handed a TBNT. The last one I interviewed at offered me a job. A problem during the medical meant I couldn't accept.
Each airline required a stamped, certified logbook from my employers.