Biologically relevant monolayer and bilayer films often consist of micron-scale high viscosity domains in a continuous low viscosity matrix. Here we show that this morphology can cause the overall monolayer fluidity to vary by orders of magnitude over a limited range of monolayer compositions. Modeling the system as a two-dimensional suspension in analogy with classic three-dimensional suspensions of hard spheres in a liquid solvent explains the rheological data with no adjustable parameters. In monolayers with ordered, highly viscous domains dispersed in a continuous low viscosity matrix, the surface viscosity increases as a power law with the area fraction of viscous domains. Changing the phase of the continuous matrix from a disordered fluid phase to a more ordered, condensed phase dramatically changes the overall monolayer viscosity. Small changes in the domain density and/or continuous matrix composition can alter the monolayer viscosity by orders of magnitude.