The $(e, e_{}^{\prime}K_{}^{+})$ experiments done at the Jefferson Laboratory provide us high-resolution structure information on various excited states of hypernuclei. Among others, the $\Lambda$ hyperon plays a unique role of inducing intershell mixing of the positive-parity and negative-parity nuclear core-excited states. In order to elucidate such new phenomena realized in hypernuclei, we have proposed an extended shell model which treats the different-parity nuclear excited states coupled with a $\Lambda$ hyperon in a consistent manner. In fact the model has been proved successful in explaining for the first time an intruder peak observed at considerably low excitation energy in the ${}_{}^{10}\mathrm{B}\,(e, e_{}^{\prime}K_{}^{+})\,{}_{\,\Lambda}^{10}\mathrm{Be}$ experiment. Here the extended shell model is applied to study novel structures of ${}_{\,\Lambda}^{11}\mathrm{B}$ and ${}_{\,\Lambda}^{11}\mathrm{Be}$. In these hypernuclear eigenstates, we also found such new states that the mixing of different-parity nuclear excited states is induced by the participation of a $\Lambda$ hyperon. The wave functions obtained in the extended model space calculations are used to predict the DWIA cross sections for the ${}_{}^{11}\mathrm{B}\,(K_{}^{-}, \pi_{}^{-})\,{}_{\,\Lambda}^{11}\mathrm{B}$ and ${}_{}^{11}\mathrm{B}\,(\gamma, K_{}^{+})\,{}_{\,\Lambda}^{11}\mathrm{Be}$ reactions, and discuss that such unique states should be possibly observed in the high-resolution reaction spectroscopy. We also evaluate the M1, E2 and E1 transtion probabilities among the excited states of the produced hypernuclei. These predictions will be useful in future spectroscopic study of hypernuclear structure when the detailed coincidence measurements of $(\pi_{}^{+}, K_{}^{+}\gamma)$ and $(K_{}^{-}, \pi_{}^{-}\gamma)$ reactions are realized in the beamline upgrade planned at the J-PARC facility.