Primordial inflation provides initial conditions for the following big-bang era that are in excellent agreement with the measured Cosmic Microwave Background (CMB) anisotropies. This requires that the field responsible for inflation (the so called inflaton) has an extremely flat potential energy. Axion / natural inflation is a very interesting class of models in which an approximate shift symmetry protects the required flatness of the potential against radiative corrections. In these models, the inflaton typically decays into gauge fields. This coupling can lead to gauge field amplification already during inflation, at scales much smaller than the CMB ones. The produced gauge field can then source gravitational waves (GW) and density perturbations, at a large enough level to generate primordial black holes (PBH). A particularly interesting case is the generation of PBH with sub-lunar masses, that can constitute the totality of the dark matter of the Universe. This phenomenon is unavoidably accompanied by a GW signal peaked in the observational band of the future laser interferometer space antenna (LISA). This GW signal has a characteristic non-Gaussian nature that is strong enough to be observed by LISA, and that is a clear smoking gun for this PBH-dark matter association.