Extreme high-energy peaked BL Lac objects (EHBLs) are an emerging class of blazars with exceptional spectral properties, e.g., enhanced emission in the hard X-ray region and a hard spectrum at very high gamma-ray energies. Currently, no suitable model exists to explain the high-energy radiation mechanism of EHBLs. Here we apply a new model of the emission process to describe the Spectral Energy Distribution (SED) of EHBLs composed of two separate components. The internal component occurs in the jet, describes synchrotron radiation and inverse Compton (IC) scattering of ultra-relativistic electrons; the two processes produce two humps of SED. The external component occurs out of the jet and considers the interaction of the proton accelerated in the jet with the cosmic microwave background (CMB), which generates electron pairs and, in turn, a cascade spectrum due to the IC scattering of those pairs with soft photons. We reproduce the SED of eight EHBLs by finding suitable values of the parameters, meanwhile, compare the gamma-ray flux with the sensitivity of Large High Altitude Air Shower Observatory (LHAASO) and verify if these objects will be observable with LHAASO. Furthermore, we discuss the influence of intergalactic magnetic fields (IGMFs) of strength 5 x 10(-18) G <= B-IG <= 5x10(-14) G and the region of proton's turbulence acceleration. Finally, we find that the model may be employed to explain the high-energy emission mechanism and predict that LHAASO may detect the high-energy data of 1ES 0347-121, 1ES 1218+304, 1ES 1959+650, 1ES 2344+514 and RGB J0710+591 in the near future.