  In the present paper, we construct spontaneously scalarized rotating black
hole solutions in dynamical Chern-Simons (dCS) gravity by following the scalar
field evolution in the decoupling limit. For the range of parameters where the
Kerr black hole becomes unstable within dCS gravity the scalar field grows
exponentially until it reaches an equilibrium configuration that is independent
of the initial perturbation. Interestingly, the $\mathbb{Z}_2$ symmetry of the
scalar field is broken and a strong maximum around only one of the rotational
axes can be observed. The black hole scalar charge is calculated for two
coupling functions suggesting that the main observations would remain
qualitatively correct even if one considers coupling functions/coupling
parameters producing large deviations from the Kerr solution beyond the
decoupling limit approximation.
