Decays of $\mathit{\Xi}_{b}^-$ and $\mathit{\Omega}_{b}^-$ baryons to $\mathit{\Lambda}_{c}^+ h^- h^{\prime -}$ final states, with $h^- h^{\prime -}$ being $\pi^-\pi^-$, $K^-\pi^-$ and $K^-K^-$ meson pairs, are searched for using data collected with the LHCb detector. The data sample studied corresponds to an integrated luminosity of $8.7\,\mathrm{fb}^{-1}$ of $pp$ collisions collected at centre-of-mass energies $\sqrt{s} = 7$, $8$ and $13\,\mathrm{Te\kern -0.1em V}$. The products of the relative branching fractions and fragmentation fractions for each signal mode, relative to the $B^- \to \mathit{\Lambda}_{c}^+ \overline{p} \pi^-$ mode, are measured, with $\mathit{\Xi}_{b}^- \to\mathit{\Lambda}_{c}^+ K^- \pi^-$, $\mathit{\Xi}_{b}^- \to\mathit{\Lambda}_{c}^+ K^- K^-$ and $\mathit{\Omega}_{b}^- \to\mathit{\Lambda}_{c}^+ K^- K^-$ decays being observed at over $5\,\sigma$ significance. The $\mathit{\Xi}_{b}^- \to\mathit{\Lambda}_{c}^+ K^- \pi^-$ mode is also used to measure the $\mathit{\Xi}_{b}^-$ production asymmetry, which is found to be consistent with zero. In addition, the $B^- \to \mathit{\Lambda}_{c}^+ \overline{p} K^-$ decay is observed for the first time, and its branching fraction is measured relative to that of the $B^- \to \mathit{\Lambda}_{c}^+ \overline{p} \pi^-$ mode.