I can't find datasheet for MJE355. I suppose that you meant MJE3055. I would try with logic level mosfet.
One reason for getting to hot is slope, as Ioannis mention, but probably isn't that.
Other:
1) What is type of transformer core, maybe frequency is to high.
2) Is that flyback topology? Maybe core is saturated.
3) Maybe emitter collector breakdown occur, I don't see any clamp on collector.
4) Neon have very low impedance when glowing, so try to put resistor in series to limit current.
etc...