Fellow with Her Maj getting a bite to eat (5)

I believe the answer is:

doner

I'm a little stuck... Click here to teach me more about this clue!

'a bite to eat' is the definition.
I can't judge whether this definition defines the answer.

'fellow with her maj' is the wordplay.
I cannot really see how this works, but
'fellow' could be 'don' (both can mean an academic) and 'don' is found in the answer.
'her maj' could be 'er' (Elizabeth Regina ie Her Majesty Queen Elizabeth) and 'er' is found in the answer.
No letters remain.
This explanation may well be incorrect...

'getting' is the link.

Can you help me to learn more?

(Other definitions for doner that I've seen before include "kebab" , "fare from Turkey" .)

I've seen this clue in The Independent.

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