Holographic particle characterization uses quantitative analysis of holographic microscopy data to precisely and rapidly measure the diameter and refractive index of individual colloidal spheres in their native media. When this technique is applied to inhomogeneous or aspherical particles, the measured diameter and refractive index represent properties of an effective sphere enclosing each particle. Effective-sphere analysis has been applied successfully to populations of fractal aggregates, yielding an overall fractal dimension for the population as a whole. Here, we demonstrate that holographic characterization also can measure the fractal dimensions of an individual fractal cluster by probing how its effective diameter and refractive index change as it undergoes rotational diffusion. This procedure probes the structure of a cluster from multiple angles and thus constitutes a form of tomography. Here we demonstrate and validate this effective-sphere interpretation of aspherical particles' holograms through experimental studies on aggregates of silica nanoparticles grown under a range of conditions.