Both hands should be used. Or whatever sound milk makes. Remember things can, and often do go awry, for instance if you get a mean tempered bovine, they can be wild. They may kick the whey out of you. Meanwhile they'll turn the milk bucket over and spill the milk.
Flies create problems when they bite.
That makes the animals switch their tails from side to side. A long tail with a wad of heavy hair on the end can be painful as it wraps around your head. Cockleburs intertwined in the long hair is a killer. What cow, in her right mind, wants cold hands wrapped around her teats. She's apt to start kicking and bucking.