Call me paranoid. I don't uncover my beer to stir. Once I hit "flameout" I play defense. Beer is exposed to sanitized surfaces/air only as much as possible. There is a small opening in my fermenter for my O2 line and stone, which gets covered with foil.
When do you cover the wort after boiling?
Immersion chiller goes in the kettle 10-15 minutes before end of boil. At flameout I put the lid on over the IC and cover any gaps with foil or a towel soaked in Starsan while the IC is running. I will occasionally lift the lid to stir with the IC, but I don't leave it uncovered, even though it would cool much faster. After cooling I drain to fermenter and aerate with oxygen immediatley prior to pitching yeast. Now that I have pumps, I can recirculate my wort with a pump while cooling, which should really speed things up. Will find out Saturday.
I recirculate with a pump and the Jamil Z return arm, with cool tap water in the winter 10 gallons of wort cools to 65F in 14.5 minutes with a 50 ft. 1/2 inch immersion chiller.