This does not mean her mother had two copies of the blue egg gene. Since blue is dominant, she only needs one blue gene to lay a green egg. But that does not matter. If your Polish/EE cross is laying a green egg, she inherited the blue one from her mother. Since her father came from a white egg, your basic assumption is right. Your cross has one white gene and one blue gene.
Since she is laying a green egg, brown is in the mix so the odds are tremendous all the eggs with the blue gene will be green even if you cross her with a rooster from a white egg breed. So I’ll use green for egg color in your case.
It may help to think about it with symbols. Some people understand it better that way. Use uppercase “O” for the dominant blue gene and lowercase “o” for the recessive white gene. Your split hen would be O,o
If you breed your cross to a rooster that has two copies of the blue egg gene, about half the offspring will get two copies of the blue egg gene and about half will get one blue and one white. All the daughters will lay a green egg but you won’t know which have two copies of the blue gene or which are split blue/white.
The rooster would be O,O. When you cross chickens they give one of the two genes they have at that point on the chromosome to their offspring, but which one is purely random. If you cross the O,o hen with the O,O rooster, all the offspring will get an O from the rooster but half will get an O from the hen and half will get an o. So about half the offspring will be O,O and half will be O,o.
I think it’s worth saying that this is random whether the hen gave an O or an o to each chick. When I say half or ¼ that is just the odds. It’s very possible if you hatch out 4 chicks, all 4 will get the O or all four might get the o. You just don’t know. You have to hatch a lot of chicks for the odds to mean anything and they really don’t mean anything with an individual chick.
If you cross her to a rooster that has one blue and one white, about ¼ of their offspring will be pure for the blue egg gene, about ½ will be split blue/white and still lay a green egg (at least the pullets. I know roosters don’t lay eggs), and about ¼ will not have any blue egg genes so will lay brown eggs.
In this case the rooster and hen would both be O,o. Each will give about half their offspring an O and half on o. About half the chicks that got an O from their father will get an O from their mother and be O,O. Half will get an o and be O,o. Half the chicks that got and o from the rooster will get an O from the hen and be O,o. Half will get an o and be o,o.
When you do the math, ¼ are O,O, half are O,o, and ¼ are o,o.
The last cross is the hen O,o and the rooster o,o or a white or brown egg. Half will be O,o and half will be o,o.
Remember that as long as one O is present, the hen will lay a green egg.