Whirlybird has just told you how to do it, although it's against future earnings from flying rather than past earnings from other sources.
Set up a company as a vehicle to train and then employ you as a pilot, and register for VAT even though you're below the threshold. Then expense your training and bill your flying work when it arrives through the company. Hey presto ! VAT on training reclaimed and training costs set against (future) earnings.
Expect a nasty call from HMRC if you never get round to invoicing though, and possibly also if you only ever invoice one company.