From the given sentence, we can infer that you can take half of the cake. So the person wants to tell that you can take one of the two halves of the cake. Either refers to one of the two given things. So option A either is correct.
Option B neither means you can't take either of two halves, but they have said you can
Option C each means you can take both the halves, but in that case, they would have said, you can take all the cake.
Option D every is used to refer to something in a group, whereas here individual pieces are referred.