  Gravitational spectroscopy - the measurement of the quasi-normal modes of a
black hole from the ringdown signal of a binary black hole coalescence - is one
of the most promising tools to test gravity in the strong-field,
large-curvature regime, but without the knowledge of the black hole
quasi-normal modes in specific cases of modified gravity theories, only null
tests of general relativity are possible. More specifically, we need to know
the modes of rotating black holes, because typical compact binary mergers lead
to black holes with large spins. In this article we compute, for the first
time, the gravitational quasi-normal modes of rotating black holes in a
modified gravity theory, up to first order in the spin. We consider
Einstein-dilaton Gauss-Bonnet gravity, one of the simplest modifications of
general relativity in the large-curvature regime. We find that the shifts in
the mode frequencies and damping times due to general relativity modifications
are significantly magnified by rotation.
