  Neutrino interactions with protons and neutrons probe their deep structure
and may reveal new physics. The higher the neutrino energy, the sharper the
probe. So far, the neutrino-nucleon ($\nu N$) cross section is known across
neutrino energies from a few hundred MeV to a few PeV. Soon, ultra-high-energy
(UHE) cosmic neutrinos, with energies above 100 PeV, could take us farther. So
far, they have evaded discovery, but upcoming UHE neutrino telescopes endeavor
to find them. We present the first detailed measurement forecasts of the UHE
$\nu N$ cross section, geared to IceCube-Gen2, one of the leading detectors
under planning. We use state-of-the-art ingredients in every stage of our
forecasts: in the UHE neutrino flux predictions, the neutrino propagation
inside Earth, the emission of neutrino-induced radio signals in the detector,
their propagation and detection, and the treatment of backgrounds. After 10
years, if at least a few tens of UHE neutrino-induced events are detected,
IceCube-Gen2 could measure the $\nu N$ cross section at center-of-mass energies
of $\sqrt{s} \approx 10-100$ TeV for the first time, with a precision
comparable to that of its theory prediction.
