I had a problem of my engine cutting out at first, but the method was slightly different...

Basically, the engine would run for 10-15 seconds then cut out.

I narrowed the problem down to an air-leak in the exhaust, so there wasn't enough backpressure to pump fuel into the engine. Maybe yours is a less severe case of the same problem, whereby when you need more fuel there isn't enough pressure to supply it?

I'd check that all your exhaust bits and bobs are on tight and undamaged...

Sorry if this seems dumb, but I'm still learning ya know :P