That's not a big cut -- I ride with stuff like that all the time. A boot would be overkill, but *if* there is damage on the inside, I would use a regular patch. Otherwise use nothing. I would not use duct tape as it's more likely to rough up your tube. Don't mess with shoe goo or filling the hole.

Having said all that, I'd probably replace a tire that had 2K miles on it. But maybe your roads are smoother so you get more mileage. BTW, almost everyone here (including myself) recommends against rotating tires. Use the search function to find out why as there are quite a few threads on the topic.