We study observables in the scattering of classical, spinning objects using the KMOC formalism. In particular, we derive formulas to higher order in spin and one loop \(\mathcal{O}({G}^{2})\) for the spin kick and momentum impulse. Our derivation method is agnostic to the choice of theory, spin representation, or special conditions, such as the spin supplementary condition (SSC); we only rely on the generic structure of long-range scattering amplitudes of non-transverse, massive spinning fields in the classical limit. We check these formulas for the case of gravity up to quadratic order in spin and agree with previous results from the eikonal formalism after imposing a SSC. This is the first derivation of its kind using KMOC and field theory to derive formulas directly relating amplitudes to observables to general order in spin without relying on a truncation in the spin expansion.
