it's better not to feed pigeons at all but if you do remove the head, neck,trachea, intestines & then deep freeze for at least 3month. coturnix quail are probably the best food package available for a long wing. Pigeons carry trichonoma.aspergillosis herpes coccidiosis,gapes a multitude of diseases to which raptors are susceptible. So on balance it's entirely up to you.
A people who would trade liberty for security will end up losing both & deserving neither.
There are so many Pigeon decease i could go on forever. The thing is worse than a Rat by a long chalk.
Many deceases are in its brain and head with others mainly in its intestines. I saw a wild Peregrine once that was rescued half its face had been eaten away by a decease carried by Pigeons, thankfully it was saved by a Raptor centre but still looks terrible. Newcastle Decease is a Killer, And your bird has very little chance of survival.
If you must feed pigeon de-gut it chop its head AND neck off and freeze it for 48 hours...Job done.
If your bird catches one get in quick and get him/her off it!
My bird recently caught one, he had 1 hour before caught a rabbit, i tossed the rabbit to one side and my bird went for it , i put the pigeon in the bin where they belong!
WHEN THE 'EAGLES' ARE SILENT THE 'PARROTS' BEGIN TO JABBER! SIR W.CHURCHILL