  We report the first measurement of the \eep three-body breakup reaction cross
sections in helium-3 ($^3$He) and tritium ($^3$H) at large momentum transfer
($\langle Q^2 \rangle \approx 1.9$ (GeV/c)$^2$) and $x_B>1$ kinematics, where
the cross section should be sensitive to quasielastic (QE) scattering from
single nucleons. The data cover missing momenta $40 \le p_{miss} \le 500$ MeV/c
that, in the QE limit with no rescattering, equals the initial momentum of the
probed nucleon. The measured cross sections are compared with state-of-the-art
ab-initio calculations. Overall good agreement, within $\pm20\%$, is observed
between data and calculations for the full $p_{miss}$ range for $^3$H and for
$100 \le p_{miss} \le 350$ MeV/c for $^3$He. Including the effects of
rescattering of the outgoing nucleon improves agreement with the data at
$p_{miss} > 250$ MeV/c and suggests contributions from charge-exchange (SCX)
rescattering. The isoscalar sum of $^3$He plus $^3$H, which is largely
insensitive to SCX, is described by calculations to within the accuracy of the
data over the entire $p_{miss}$ range. This validates current models of the
ground state of the three-nucleon system up to very high initial nucleon
momenta of $500$ MeV/c.
