  We show a tight-binding model for double perovskite monolayers grown in the
{001} crystallographic direction can exhibit higher Chern numbers for the
lowest band of the model dispersion in the absence of an applied magnetic
field, including $\pm2$, $\pm4$, $\pm6$, and $\pm8$, for model parameters and
terms relevant to Sr$_2$CrWO$_6$ (SCWO). We further show it is possible to tune
between these different topological phases by applying a weak in-plane magnetic
field. These results may be important for experimentally realizing exotic band
topologies -- and even quantum anomalous Hall insulators with higher Chern
numbers -- based on this physics.
