Check out this thread on the subject:
https://www.backyardchickens.com/forum/viewtopic.php?pid=2737058
EE is not a breed - not "pure" - so you CAN get an EE by breeding to a roo of another breed and the resulting chicks will be EE (mutts). It's when you're breeding for egg color that you must pay attention to the eggs the parents came from.
Here's an enlightening treatise from Kev (from the link above) that you may find helpful:
"EE isn't a breed so there is zero standard so they can look like anything. However so many EE have a beard and green legs a lot of people just expect EE to have beard and green legs, as if that's a "breed trait".. but the again, EE are not a breed.
You CAN cross some of your EE with whatever breed or mixed roo that strikes your fancy and still sell the chicks as EE anyways. Remember, no standard on 'em..
....based on the fact EE gene pool is so wide and mixed that there are a lot of EE that lay tan eggs, meaning they do not have the blue egg gene. So, even if an EE hen lays blue or green eggs does not mean she is automatically pure for the blue eggshell gene. In other words, do not be surprised if some daughters out of EE, however you bred them, lay tan or brown eggs instead of blue or green."
I suppose in a lot fewer words, anything goes with EE.. lol.. Just breed them however you like.. EE roo, another breed, or a mixed roo over the EE hens will all just be fine. But, if your goal is more blue/green egg laying daughters it is better to use the EE roo as roosters can have the blue egg gene too. So if your EE roo happens to have it or even is pure for the blue egg gene, then you will get a lot more blue/green egg laying pullets out of him and these hens."
Good luck!