In this study, we propose a new hyperbolic model capable to capture the wave breaking phenomenon. The modelling of breaking waves is obtained by the depth- averaging method of Large-Eddy Simulations (LES) where the small scale turbulence is modeled by a turbulent viscosity, whereas the large scales are taken into account in the model by an anisotropic tensor variable called enstrophy. The hyperbolic structure is derived by replacing the depth-averaged non-hydrostatic pressure with an additional variable. The hyperbolisation of the equations is based on taking into account the finite character of the sound speed and introducing acoustic energy into the system. The resulting model can be viewed as a hyperbolic approximation of the Serre-Green-Naghdi (SGN) equations. Additionally, it has asymptotic dispersive properties to SGN equations as approaching to infinite sound speed. The treatment of breaking wave is to use the so-called switching method which certain terms describing the energy dissipation are activated once the wave breaks. So, we also present a more-robust breaking criteria on which only depends the local variables.