Unfortunately "EE" doesn't mean anything. They are not a breed so there is no breed standard. We can't even agree on a definition of what an EE is. As you just pointed out, an EE hen doesn't even need to lay a blue or green egg to be called an EE.
The way the egg shell color genetics work one gene pair determines if the base color is going to be blue or white. The Blue gene is dominant so if either one or both of those genes at that gene pair is the Blue gene the egg shell base color will be blue. If neither is Blue it defaults to white. So that gene pair sets the base color.
There are many different gene pairs that affect the brown color. Some are dominant, some are recessive. Some only act if another certain gene is present. One bleaches brown so it disappears. One is sex linked. There are a whole lot of different possible combinations, that's why you can get so many different shades of brown. A white egg becomes brown if a hen deposits brome on a white egg. A blue egg becomes green if a hen deposits brown on blue. This might explain it.
Blue + no brown = blue
Blue + brown = green
White + no brown - white
White + brown = brown
Both parents contribute equally to those blue/white base color genetics. The genes at that gene pair are handed down randomly. I think graphics can help us understand. The symbol for the dominant Blue gene is uppercase "O". The non-dominant Not-Blue symbol is lower case "o". If either parent is O/O at that gene pair then one will hand down an "O" so the pullet will lay a base blue egg. If both parents are o/o the pullet will lay a base white egg. If one parent is o/o while the other is O/o, the pullet might lay a base blue egg or she might lay a base white egg. The odds are 50-50 each way. If both parents or O/o the odds are 75% that the pullet will lay a base blue egg and 25% that she will lay a base white egg.
Since your EE laid a brown egg she has no blue to contribute, she is o/o. But the Legbar rooster should be pure for the blue egg shell gene, or O/O. That means a pullet from that crossing should be O/o and lay a base blue egg. What shade of blue or green will depend on what brown gets contributed.
This is probably clear as mud. But then the definition of an EE is also clear as mud because it can mean different things to different people. But the pullets from your cross should lay a base blue egg. That will come from the father, not the mother.