Soft-solid foods show a progressive transition from a viscoelastic solid state to a flowing fluid state when subjected to a large load. The engineering properties and sensory texture of soft-solid foods depend strongly on the rheological properties that characterize this fluidization. In this paper we use Large Amplitude Oscillatory Shear (LAOS) rheometry to quantify the texture of emulsion-filled food gels in terms of measurable material properties. We provide unambiguous rheological definitions for the firmness, rubberiness, softening and fluidization of soft-solid food gels. We propose a new measure for the load-induced solid-fluid transition, the fluidizing ratio, which quantifies the progression of damage and the degree of plastic flow in the soft-solid gel. We use another dimensionless measure, the thickening ratio, to reveal and characterize the resulting sequence of flow regimes. We use our rheological definitions to quantify the texture of zero-fat, low-fat and full-fat semi-hard cheese respectively. Our data provides evidence that the rate of two physical processes, microcrack nucleation and microcrack propagation, are controlled by the amount of fat emulsion in the gel and govern the rubberiness and brittleness of semi-hard cheese. By translating texture terminology into quantitative material properties measured using Large Amplitude Oscillatory Shear, we augment the capabilities of LAOS as an analytical tool for structure-texture engineering of soft-solid food gels.