  Using gravitational wave observations to search for deviations from general
relativity in the strong-gravity regime has become an important research
direction. Chern Simons (CS) gravity is one of the most frequently studied
parity-violating models of strong gravity. It is known that the Kerr black-hole
is not a solution for CS gravity. At the same time, the only rotating solution
available in the literature for dynamical CS (dCS) gravity is the slow-rotating
case most accurately known to quadratic order in spin. In this work, for the
slow-rotating case (accurate to first order in spin), we derive the linear
perturbation equations governing the metric and the dCS field accurate to
linear order in spin and quadratic order in the CS coupling parameter
($\alpha$) and obtain the quasi-normal mode (QNM) frequencies. After confirming
the recent results of Wagle et al. (2021), we find an additional contribution
to the eigenfrequency correction at the leading perturbative order of
$\alpha^2$. Unlike Wagle et al., we also find corrections to frequencies in the
polar sector. We compute these extra corrections by evaluating the expectation
values of the perturbative potential on unperturbed QNM wavefunctions along a
contour deformed into the complex-$r$ plane. For $\alpha=0.1 M^2$, we obtain
the ratio of the imaginary parts of the dCS correction to the GR correction in
the first QNM frequency (in the polar sector) to be $0.263$ implying
significant change. For the $(2,2)-$mode, the dCS corrections make the
imaginary part of the first QNM of the fundamental mode less negative, thereby
decreasing the decay rate. Our results, along with future gravitational wave
observations, can be used to test for dCS gravity and further constrain the CS
coupling parameters. [abridged]
