I have an idea.. you say you don't remember what he looks like until you see him?? This is an interesting and important point.
Why don't we get a collection of pictures of owls, including cartoon owls, including the owl you're afraid of, in a photo-fit line-up of about 20 arranged 5x4. You point out which one you're afraid of.
This will mean that your brain must compare and analyse the images in order to identify and isolate the characteristics of the feared character. It will interrupt your normal panic response for a moment... A moment in which your brain might choose to break the phobia, as it associates it with the other owls you're seeing which you know are harmless.
Anyone volunteer to make the photo line-up?
Jasmine