You probably want to use a wrought iron pan for that. They can be really cheap (starting from less than 10€), are indestructible (they don’t break if you drop them) and can be much thinner than cast iron. They are better for eggs, where you want to change the temperature during frying very quickly.
Yes, a good Matfer or Solidteknics carbon steel / wrought iron / blue iron pan is fantastic for eggs, omelettes, and crepes. Matfer as a French brand even has specific crepe and omelette pans made this way.
The only downside to a Matfer, or any other carbon steel pan, is that you need to be prepared to season it yourself. They do not come pre-seasoned like most cast iron pans and it is a good bit of work. Having said that, I absolutely love the one I have and plan on buying at least one more in the future.
