TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Process: The process comprises: mixing fatty acid alkyl ester with polar solvent and non-polar solvent by stirring to obtain the mixture, and then decanting the mixture until a heavy phase enriched in monoglycerides, and a light phase including purified fatty acid alkyl ester. The process comprises at least two extraction stages, preferably at least five extraction stages. Preferred Components: The lower alcohol is methanol (preferred), ethanol, isopropanol, n-propanol, butanol, isobutanol and/or 2-ethylhexanol. The polar solvent comprises lower alcohol (99.9-70%) and water (0.1-30%). The non-polar solvent is hexane (preferred), heptane, benzene, bicyclohexyl, cyclohexane, decalin, decane, hexane, kerosene, methylcyclohexane, Texsolve S (RTM: Mineral spirits) or Texsolve S-66 (RTM: Mineral spirits), naphtha, Shellite (RTM: Mixture of paraffins and naphthenes), tetradecane, supercritical carbon dioxide, pressurized butane or propane, and/or natural solvents. The fatty acid alkyl ester includes at least one ester comprising ester of oils of rapeseed, safflower, sunflower, nasturtium, mustard seed, olive, sesame, soybean, corn, peanut, walnut, hazelnut, avocado, grape, cotton seed, rice, babassu, castor, palm, palm kernel, rapeseed, lupine, Jatropha, coconut, flax, evening primrose, jojoba, Camelina, algae, tallow, such as beef tallow or pork, chicken fat, lard, fish, milk fat, shea butter, biodiesel, cooking oil, frying oil, miscella, derivatives of these oils, including hydrogenated derivatives, conjugated derivatives and/or their oil cuts. The fatty acid alkyl ester cut comprises at least 90 wt.% of fatty acid methyl ester. The polar solvent is introduced against counter-current of fatty acid alkyl ester cut for obtaining the production of a heavy phase comprising purified fatty acid alkyl ester, and a light phase enriched in monoglycerides. The weight ratio of polar solvent/fatty acid alkyl ester is 1/5-5/1, preferably 1/4-1/1.