Pulsar timing arrays are a prime tool to study unexplored astrophysical regimes with gravitational waves. Here, we show that the detection of gravitational radiation from individually resolvable supermassive black hole binary systems can yield direct information about the masses and spins of the black holes, provided that the gravitational-wave-induced timing fluctuations both at the pulsar and at Earth are detected. This in turn provides a map of the nonlinear dynamics of the gravitational field and a new avenue to tackle open problems in astrophysics connected to the formation and evolution of supermassive black holes. We discuss the potential, the challenges, and the limitations of these observations.