  The gravitational waves emitted in the ringdown phase of binary black-hole
coalescence are a unique probe of strong gravity. Understanding how deviations
from general relativity affect the ringdown phase of black holes, however, is
extremely challenging, as it requires solving highly-coupled and sometimes
higher-order partial differential equations. We here extend a novel approach,
\textit{Metric pErTuRbations wIth speCtral methodS} (METRICS), to study the
metric perturbations and the quasinormal mode frequencies of ringing black
holes in modified gravity. We first derive the asymptotic behavior of metric
perturbations at the event horizon and spatial infinity for rotating black
holes beyond general relativity. We then extend the eigenvalue-perturbation
theory approach of METRICS to allow us to compute the leading-order modified
gravity corrections to the quasinormal-mode frequencies and metric
perturbations. We apply METRICS to rotating black holes in scalar-Gauss-Bonnet
gravity. Without decoupling or simplifying the linearized field equations in
this theory, we compute the leading-order corrections to the quasinormal
frequencies of the axial and polar perturbations of the $nlm = 022$, 021 and
033 modes of black holes of $a \leq 0.85$. The numerical accuracy of the
METRICS frequencies is $\leq 10^{-5}$ for $a \leq 0.6$, $\lesssim 10^{-4}$ for
$0.6 < a \leq 0.7 $, and $\lesssim 10^{-2}$ for $0.7 < a \leq 0.85 $ for all
modes studied. We fit the frequencies as a polynomial in spin, whose
coefficients (up to second order in spin) are consistent with those obtained in
previous slow-rotating approximations. These results are the first accurate
computations of the gravitational quasinormal-mode frequencies of rapidly
rotating black holes (of $a \sim 0.85$) in scalar-Gauss-Bonnet gravity.
