We consider a particular integral transform related to the X-ray transform that was inspired by the recent work of Kenig and Salo on the anisotropic inverse conductivity problem for a particular class of Riemannian manifolds. Given a subset E of the boundary of a domain such that the complement of E (in the boundary) is contained in a union of hyperplanes, we measure the attenuation of all interior broken rays starting and ending at E with the standard optical reflection rule applied to the reflecting surface. By localizing the measurement operator around broken rays which reflect off of a fixed sequence of flat hyperplanes, one can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann for the ordinary ray transform by means of a local path unfolding. I will discuss this approach and how injectivity and stability of the transform can occur for certain choices of E.