  We construct a new Yang-Mills Lagrangian based on a notion of minimal
coupling that incorporates classical spin effects. The construction relies on
the introduction of a new covariant derivative, which we name "classical spin
covariant derivative", that is compatible with the three-point interaction of
the $\sqrt{\mathrm{Kerr}}$ solution with the gauge field. The resulting
Lagrangian, beside the correct three-point coupling, predicts a unique choice
for contact terms and therefore it can be used to compute higher-point
amplitudes such as the Compton, unaffected by spurious poles. Using double-copy
techniques we use this theory to extract gravity amplitudes and observables
that are relevant to describe Kerr binary dynamics to all orders in the spin.
In particular, we compute the 2PM ($\mathcal{O}(G^2_N)$) $2\rightarrow 2$
scattering amplitude between two classically spinning objects to all orders in
the spin and use it to extract the 2PM scattering angle.
