  Soon, a new generation of neutrino telescopes, presently under planning, will
target the discovery of ultra-high-energy (UHE) neutrinos of cosmic origin,
with energies higher than 100 PeV, that promise unique insight into
astrophysics and particle physics. Yet, predictions of the UHE neutrino flux
and interaction cross section -- whose measurement is co-dependent -- are laden
with significant uncertainty that, if unaddressed, could misrepresent the
capabilities to measure one or the other. To address this, we advocate for the
joint measurement of the UHE neutrino spectrum and neutrino-nucleon cross
section, including of their energy dependence, without assuming prior knowledge
of either. We illustrate our methods by adopting empirical parametrizations of
the neutrino spectrum, in forecasts geared to the planned radio array of the
IceCube-Gen2 neutrino telescope. We warn against using simple parametrizations
-- a simple power law or one augmented with an exponential cut-off -- that
might fail to capture features of the spectrum that are commonplace in the
predictions. We argue instead for the use of flexible parametrizations -- a
piecewise power law or an interpolating polynomial -- that ensure accuracy. We
report loose design targets for the detector energy and angular resolution that
are compatible with those under present consideration.
