  We study free, capped and encapsulated bilayer jacutingaite Pt$_2$HgSe$_3$
from first principles. While the free standing bilayer is a large gap trivial
insulator, we find that the encapsulated structure has a small trivial gap due
to the competition between sublattice symmetry breaking and
sublattice-dependent next-nearest-neighbor hopping. Upon the application of a
small perpendicular electric field, the encapsulated bilayer undergoes a
topological transition towards a quantum spin Hall insulator. We find that this
topological transition can be qualitatively understood by modeling the two
layers as uncoupled and described by an imbalanced Kane-Mele model that takes
into account the sublattice imbalance and the corresponding inversion-symmetry
breaking in each layer. Within this picture, bilayer jacutingaite undergoes a
transition from a 0+0 state, where each layer is trivial, to a 0+1 state, where
an unusual topological state relying on Rashba-like spin orbit coupling emerges
in only one of the layers.
