I have a 15 day entry permit which I received at Mae Sai because I left my passport with the Burmese authorities rather than the Thai authorities when I went for a day trip (voided my tourist visa). The permit expires 15th of this month and I intend to go up tomorrow (the 13th) to sort something out.

I have a flight home booked for 17th of NEXT month. What options do I have to minimise the number of trips I have to make to Mae Sai? I know I can get 15 days, which would mean another 3 trips (or 2 trips if I risk a few days overstay). Is it possible for me get 30 days? Also, I don't know whether to handle this at the immigration office in Mae Sai or the border.

You can not get 30 days. You can not get more than 7 day extension. You need to cross border and return for new 15 day stays unless you are willing to fly somewhere and return for 30 days or visit Vientiane for a tourist visa.