The phrase "to coat the back of a spoon" means to dissolve or disperse something in a liquid, so that it forms a thin film on the surface of the liquid. This is often done by stirring the mixture with a spoon while holding it upside down over another container, allowing the liquid to drain off and leaving behind only the solid particles suspended in the liquid. The phrase can also refer to adding an ingredient or substance to a liquid so that it forms a thin layer on the surface of the liquid.