You don't recover from a speech impediment - You overcome it.
'It took me ages to recover from a serious bout of shingles.'
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Over/overly are different. They mean excessive/excessively.
'My wife is over-generous/overly-generous with her money; she gives too much away.'
So you can see it doesn't make sense to say 'over-recover from an illness'. It would mean 'I am excessively healthy.'
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