  We suggest an extension to isospin asymmetric matter of the quarkyonic model
from McLerran and Reddy. This extension allows us to construct the
$\beta$-equilibrium between quarks, nucleons and leptons. The concept of the
quarkyonic matter originates from the large number of color limit for which
nucleons are the correct degrees of freedom near the Fermi surface --
reflecting the confining forces -- while deep inside the Fermi sea quarks
naturally appear. In isospin asymmetric matter, we suggest that this new
concept can be implemented within a global isoscalar relation between the shell
gaps differentiating the nucleon and the quark sectors. In addition, we impose
the conservation of the isospin-flavor asymmetry in the nucleon and the quark
phases. Within this model, several quarkyonic stars are constructed on top of
the SLy4 model for the nucleon sector, producing a bump in the sound speed,
which implies that quarkyonic stars are systematically bigger and have a larger
maximum mass than the associated neutron stars. They also predict lower proton
fraction at $\beta$-equilibrium, which potentially quenches fast cooling in
massive compact stars.
