  Rapidly rotating black hole solutions in theories beyond general relativity
play a key role in experimental gravity, as they allow us to compute
observables in extreme spacetimes that deviate from the predictions of general
relativity. Such solutions are often difficult to find in
beyond-general-relativity theories due to the inclusion of additional fields
that couple to the metric non-linearly and non-minimally. In this paper, we
consider rotating black hole solutions in one such theory, dynamical
Chern-Simons gravity, where the Einstein-Hilbert action is modified by the
introduction of a dynamical scalar field that couples to the metric through the
Pontryagin density. We treat dynamical Chern-Simons gravity as an effective
field theory and work in the decoupling limit, where corrections are treated as
small perturbations from general relativity. We perturb about the
maximally-rotating Kerr solution, the so-called extremal limit, and develop
mathematical insight into the analysis techniques needed to construct solutions
for generic spin. First we find closed-form, analytic expressions for the
extremal scalar field, and then determine the trace of the metric perturbation,
giving both in terms of Legendre decompositions. Retaining only the first three
and four modes in the Legendre representation of the scalar field and the
trace, respectively, suffices to ensure a fidelity of over 99% relative to full
numerical solutions. The leading-order mode in the Legendre expansion of the
trace of the metric perturbation contains a logarithmic divergence at the
extremal Kerr horizon, which is likely to be unimportant as it occurs inside
the perturbed dynamical Chern-Simons horizon. The techniques employed here
should enable the construction of analytic, closed-form expressions for the
scalar field and metric perturbations on a background with arbitrary rotation.
