Classical interactions of spinning black holes (BHs) can be usefully described by certain limits of certain quantum scattering amplitudes, for massive higher-spin fields "minimally coupled" to gravity.  Tree-level interactions between two BHs are determined by 3-point amplitudes (BH in, BH out, meeting one graviton) which are arguably well understood -- on the quantum side, for arbitrary-spin massive particles; classically, to all orders in the BHs' spins.  Two-BH (conservative) interactions at the classical 1-loop level (and other processes) are argued to be determined by the additional 4-point "Compton" amplitude (BH in, BH out, meeting two gravitons), which is more poorly understood -- on the quantum side, up to spin-2 (or 5/2?); classically, up to 4th (or 5th?) order in the BH spin.  In this talk, we'll discuss efforts to better understand the BH-graviton Compton amplitude for higher spins, and its applications, including directly interpreting its classical limit as the outgoing amplitude for an incoming gravitational plane wave scattering off a Kerr BH