I 2nd the barrier creams. When I worked for the railroad lots of equipment was greasy/oily.I liked the barrier creams because they allow you to work without gloves and it was quite amazing how easy it was to wash away the grime when it came time to clean up.
Another trick worth trying (I use it after working on bicycle chains, have no idea how it will work for heavy dairy grease) is to use some nice skin compatible oil as a solvent for removing the grease. You can then use your ordinary soap to remove the oil, or you can simply scrape the oil off. Various vegetable oils, 'baby oil', oil based moisturizers, massage oil, etc. are all potentially useable in this fashion.
Short hair, hat and hand cleaner gets my vote. I occasionally worked on pre-WWII railway equipment and you can guess the amount of grease caked with dirt.