  We introduce a new subclass of Fano varieties (Casagrande-Druel varieties),
that are $n$-dimensional varieties constructed from Fano double covers of
dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable
if the double cover and its base space are K-polystable. We prove this for
smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties
constructed from double covers of $\mathbb{P}^{n-1}$ ramified over smooth
hypersurfaces of degree $2d$ with $n>d>\frac{n}{2}>1$. As an application, we
describe the connected components of the K-moduli space parametrizing
smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the
Mori-Mukai classification.
