After researching different ways to cook an egg, I saw that placing it in a 10-minute ice bath was supposed to make the egg contract inside the shell, making it easier to peel. In this case, though, the ice bath didn't seem to make too much of a difference. Both eggs were difficult to peel and pieces of the shells clung to the whites for dear life.
The egg that was put in the ice bath was much runnier than its non-iced counterpart. That makes sense, though, since the ice water stops the cooking process.
In the case of the ice-submerged egg, the whites were runny, which isn't ideal. While they weren't completely cooked through, they were mostly solid in the egg that didn't receive an ice bath.
I couldn't fully bite into these as they were, well, mostly raw. But, if I let them sit out of the water and cook in their own heat just a few minutes longer, the white would have continued to cook leaving me with what would resemble a poached egg.