As far as I've seen, Kohanim who are about to bless the congregation first untie their shoes (if necessary), then have their hands washed, then remove their shoes, and finally recite the blessing. As I understand it, the shoe removal must be done without touching their shoes, so that the kohanim need not wash their hands again (because they've touched their shoes) before reciting the blessing.
My kid noticed this and wondered why the kohanim don't remove their shoes before washing their hands. That, after all, would enable them to use their hands to remove their shoes, which is easier. I had no answer to this question and turn to you.