The first observations of ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{*}(2007{)}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}_{s}^{0}\ensuremath{\rightarrow}{\overline{D}}^{*}(2007{)}^{0}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ decays are presented, and their branching fractions relative to that of the ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{*}(2007{)}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay are reported. These modes can potentially be used to investigate the spectroscopy of charm and charm-strange resonances and to determine the angle $\ensuremath{\gamma}$ of the Cabibbo-Kobayashi-Maskawa unitarity triangle. It is also important to understand them as a source of potential background in determinations of $\ensuremath{\gamma}$ from ${B}^{+}\ensuremath{\rightarrow}D{K}^{+}$ and ${B}^{0}\ensuremath{\rightarrow}D{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays. The analysis is based on a sample corresponding to an integrated luminosity of $5.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collision data at 13 TeV center-of-mass energy recorded with the LHCb detector. The ${\overline{D}}^{*}(2007{)}^{0}$ mesons are fully reconstructed in the ${\overline{D}}^{0}{\ensuremath{\pi}}^{0}$ and ${\overline{D}}^{0}\ensuremath{\gamma}$ channels with the ${\overline{D}}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay. A novel weighting method is used to subtract background while simultaneously applying an event-by-event efficiency correction to account for resonant structures in the decays.