An EE is a mix. It can look like anything, be any color or pattern, and generally has no defining characteristics. Even the pea comb is not a given. Having no idea what genes are present in your EE, I can't tell you what that cross will look like.
The black in a BA dominates over a lot of things, so the odds are that the offspring will have black on them. But that is not a given. And it is quite possible the chicks will not all look the same. The EE could easily be split for certain genes that would change the appearance of the individual offspring.
If the EE is pure for the blue egg gene, all hens from this cross would lay green eggs. If the EE is split for the blue egg gene, about half the hens would lay green eggs and the rest would be brown eggs. If the EE does not have any blue egg genes, all eggs will be some version of brown.
Not a lot of help I know, but I don't see any clear answer. It could make a nice chicken though. I like Black Australorp.
Eight colors are officially recognized for poultry shows by the American Poultry Association: Black, Blue, Blue Wheaten, Brown Red, Buff, Silver, Wheaten and White. There are several project colors, including Lavender. Which do you have?