  Electrons in quantum materials exhibiting coexistence of dispersionless
(flat) bands piercing dispersive (steep) bands can give rise to strongly
correlated phenomena, and are associated with unconventional superconductivity.
It is known that in twisted trilayer graphene steep Dirac cones can coexist
with band flattening, but the phenomenon is not stable under layer
misalignments. Here we show that such a twisted sandwiched graphene (TSWG) -- a
three-layer van der Waals heterostructure with a twisted middle layer -- can
have very stable flat bands coexisting with Dirac cones near the Fermi energy
when twisted to 1.5 degrees. These flat bands require a specific high-symmetry
stacking order, and our atomistic calculations predict that TSWG always relaxes
to it. Additionally, with external fields, we can control the relative energy
offset between the Dirac cone vertex and the flat bands. Our work establishes
twisted sandwiched graphene as a new platform for research into strongly
interacting phases, and topological transport beyond Dirac and Weyl semimetals.
