  The heavy fermion paramagnet UTe$_2$ exhibits numerous characteristics of
spin-triplet superconductivity. Efforts to understand the microscopic details
of this exotic superconductivity have been impeded by uncertainty regarding the
underlying electronic structure. Here we directly probe the Fermi surface of
UTe$_2$ by measuring magnetic quantum oscillations in pristine quality
crystals. We find an angular profile of quantum oscillatory frequency and
amplitude that is characteristic of a quasi-2D Fermi surface, which we find is
well described by two cylindrical Fermi sheets of electron- and hole-type
respectively. Additionally, we find that both cylindrical Fermi sheets possess
considerable undulation but negligible small-scale corrugation, which may allow
for their near-nesting and therefore promote magnetic fluctuations that enhance
the triplet pairing mechanism. Importantly, we find no evidence for the
presence of any 3D Fermi surface sections. Our results place strong constraints
on the possible symmetry of the superconducting order parameter in UTe$_2$.
