I do eggs on my Lodge griddle. Hit it with a light film of Pam or some other non-stick cooking spray before you crack your eggs onto the surface. Let them brown a bit on the bottom, flip for over easy or medium.
After scraping an rinsing, I dry it in the oven, then rub a bit of oil on the cooking surface, then let cool.
The trouble with eggs is that they have a very high lecithin content, which emulsifies the seasoning right off the pan. “Emulsify” is a term used to describe mixing oil and water... it’s pretty bad for the surface of a pan.