Rheo-dielectric spectroscopy is employed to investigate the effect of external shear on Debye-like relaxation of a model monohydroxy alcohol, i.e., the 2-ethyl-1-hexanol (2E1H). Shear deformation leads to strong acceleration in the structural relaxation, the Debye relaxation, and the terminal relaxation of 2E1H. Moreover, the shear-induced reduction in structural relaxation time, τ_{α}, scales quadratically with that of Debye time, τ_{D}, and the terminal flow time, τ_{f}, suggesting a relationship of τ_{D}^{2}∼τ_{α}. Further analyses reveal τ_{D}^{2}/τ_{α} of 2E1H follows Arrhenius temperature dependence that applies remarkably well to many other monohydroxy alcohols with different molecular sizes, architectures, and alcohol types. These results cannot be understood by the prevailing transient chain model, and suggest a H-bonding breakage facilitated sub-supramolecular reorientation as the origin of Debye relaxation of monohydroxy alcohols, akin to the molecular mechanism for the terminal relaxation of unentangled "living" polymers.