This is really hard to describe but after cooking a turkey, i go to cut off the individual breasts to notice on the thicker end of the breast there is no meat, but under the skin rather a white mushy mass that doesnt really hold together. Granted i cut the turkey when cold. This doesnt seeme right i think it could be a tumor or something but yet seemes to be even on each breast.
I suspect it is the paper bag which contains the giblets . Did you remove it when rinsing out the bird.
Did you buy a "Butter Ball" brand turkey or something similar that has been "injected" with marinade or butter? If so, this could be a pocket of that marinade. They inject stuff in to birds all the time to "make them more juicy". Would recommend an all natural, no additives bird next time.