A process of making cheese, e.g., a mozzarella variety cheese, is disclosed. It comprises the following steps: a) preparing a cheese curd; b) grinding the curd while in admixture with (i) an aqueous solution of at least one cheese emulsifying salt and (ii) at least one GRAS food additive in the form of a comminuted solid, to obtain an emulsifier/additive-impregnated ground curd; and c) converting the emulsifier/additive-impregnated curd into cheese either by (i) heating, kneading, and stretching the emulsifier/additive-impregnated ground curd to obtain a homogeneous mass of cheese, or (ii) pressing the emulsifier/additive-impregnated ground curd to obtain a homogeneous mass of cheese. Examples of suitable GRAS food additives include gums, stabilizers, dairy solids (e.g., non-fat dry milk or whey protein concentrate), cheese powders, non-dairy protein isolates, sodium chloride, native or modified food starches, colorants, and flavorants. The process is particularly useful when a proteinaceous GRAS food additive is used.