Composite hydrogels are ubiquitous structural components of biological materials, biomedical devices, foods, and consumer products. Though their non-linear mechanical properties are understood to be critical to many of their respective applications, a detailed characterization of these non-linear properties has remained challenging. To address this issue, we introduce a technique to independently resolve the non-linear elasticity and plasticity of soft systems through large-amplitude oscillatory rheology. We demonstrate this technique on particle-filled biopolymer hydrogels as well as biological tissues, and elucidate how their non-linear mechanical properties are driven by key composite properties such as polymer chain elasticity, filler concentration, and filler-polymer interactions. These findings establish key structure-property relationships that underlie the non-linear mechanics of soft composite gels and tissues, and enable their rational design for biological and technological applications.