At the sub-stellar boundary, signatures of magnetic fields begin to manifest at radio wavelengths, analogous to the auroral emission of the magnetised solar system planets. This emission provides a singular avenue for measuring magnetic fields at planetary scales in extrasolar systems. So far, exoplanets have eluded detection at radio wavelengths. However, ultracool dwarfs (UCDs), their higher mass counterparts, have been detected for over two decades in the radio. Given their similar characteristics to massive exoplanets, UCDs are ideal targets to bridge our understanding of magnetic field generation from stars to planets. In this work, we develop a new tomographic technique for inverting both the viewing angle and large-scale magnetic field structure of UCDs from observations of coherent radio bursts. We apply our methodology to the nearby T8 dwarf WISE J062309.94-045624.6 (J0623) which was recently detected at radio wavelengths, and show that it is likely viewed pole-on. We also find that J0623's rotation and magnetic axes are misaligned significantly, reminiscent of Uranus and Neptune, and show that it may be undergoing a magnetic cycle with a period exceeding 6 months in duration. These findings demonstrate that our method is a robust new tool for studying magnetic fields on planetary-mass objects. With the advent of next-generation low-frequency radio facilities, the methods presented here could facilitate the characterisation of exoplanetary magnetospheres for the first time.