Crossword Genius

Bed shortly - sleep will be fitful having eaten a cheese (3,5)

Ross

I believe the answer is:

bel paese

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

'cheese' is the definition.
(I've seen this before)

'bed shortly sleep will be fitful having eaten a' is the wordplay.
I cannot quite see how this works, but
'be' is found in the answer.
'a' is present in the answer.
an anagram of 'sleep' is 'lpese' which is present in the remaining letters.
No letters remain.
This explanation may well be incorrect...

Can you help me to learn more?

(Other definitions for bel paese that I've seen before include "Cheese from the beautiful country" , "Creamy Italian cheese" , "Be asleep (anag) - Italian cheese" , "Mild semi-soft Italian cheese" .)

I've seen this clue in The Independent.
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