We investigate the electronic structure and the optical absorption onset of close-to-30 degrees twisted hexagonal boron nitride bilayers. Our study is carried out with a purposely developed tight-binding model validated against density functional theory simulations. We demonstrate that approaching 30∘ (quasicrystal limit), all bilayers sharing the same moiré supercell develop identical band structures, irrespective of their stacking sequence. This band structure features a bundle of flat bands lying slightly above the bottom conduction state which is responsible for an intense peak at the onset of the absorption spectrum. These results suggest the presence of strong, stable and stacking-independent excitons in boron nitride 30∘-twisted bilayers. By carefully analyzing the electronic structure and its spatial distribution, we elucidate the origin of these states as moiré-induced K-valley scattering due to interlayer B−B coupling. We take advantage of the physical transparency of the tight-binding parameters to derive a simple triangular model based on the B sublattice that accurately describes the emergence of the bundle. As our conclusions very general, we predict that a similar bundle should emerge in other close-to-30∘ bilayers, such as transition metal dichalcogenides, shedding light on the unique potential of two-dimensional materials.