Well you don't necessarily need an electrician, but you definitely need a new ceiling rose. The skirt on that lampholer is super burnt.
You probably only need to replace the lamp drop.
If you're confident, turn the power off, then unscrew the cover, undo the blue and brown cables that that white flex is connected, and replace with the new one.
You could probably just replace the skirt, for it to be that burnt, it would imply the cables have been getting warm too. Put a lower wattage lamp in next time (or even better, LED)
Sometimes when a bulb filament fails a gas plasma is formed (the gas ionises) . This causes a high current to flow in that circuit, usually sufficient to blow the lighting fuse or trip he lighting breaker. The plasma can also produce sufficient mechanical shock to either shatter the glass or blow the glass from the socket.
The "cooking" on the shroud wasn't caused by the plasma, it very much looks like overheating from the bulb's normal use.
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