We observe a resonant transition in the microwave absorption of thin thermally deposited Au nanoparticle films near the geometrical percolation transition Pc where the films exhibit a 'fractal' heterogeneous geometry. Absorption of incident microwave radiation increases sharply near p, consistent with effective medium theory predictions. Both the theory and our experiments indicate that the hierarchical structure of these films makes their absorption insensitive to the microwave radiation wavelength lambda, so that this singular absorption of microwave radiation is observed over a broad frequency range between 100 MHz and 20 GHz. The interaction of electromagnetic radiation with randomly distributed conductive scattering particles gives rise to localized resonant modes, and our measurements indicate that this adsorption process is significantly enhanced for microwaves in comparison to ordinary light. In particular, above the percolation transition a portion of the injected microwave power is stored within the film until dissipated. Finally, we find that the measured surface conductivity can be quantitatively described at all Au concentrations by generalized effective medium theory, where the fitted conductivity percolation exponents and p(c) itself are consistent with known two-dimensional estimates. Our results demonstrate that microwave measurements provide a powerful means of remotely measuring the electromagnetic properties of highly heterogeneous conducting films, enabling purposeful engineering of the electromagnetic properties of thin films in the microwave frequency range through fabrication of 'disordered' films of conducting particles such as metal nanoparticles or carbon nanotubes.