Probably by snapping the neck. I don't have any experience with minks, but with small animals like rabbits, that's the best option. They usually don't even have time to process what's happening to them, much less suffer because of it.
Absolutely correct Thejovialinquisitor. The process is to catch the mink, chicken, magpie, pidgeon tightly by the neck with the left hand, put your right hand under the chin. and force the head back and forward. Death is immediate. I have killed thousands and not one avoided instant death. In seconds this can be done and as you say it is over before the crature knows what is happening. I know some readers will find this upsetting, but in a world where billions of creatures are killed in a cruel manner each year, I can assure them my mink were killed by an animal lover.
Compare this with the time I passed by a neighbour trying to kill a chicken. I saw her place the handle of a brush over its neck on the ground and standing on each side of it. I could hear the hen swaking. I stopped the car, grabbed the hen and put it away in two seconds. I told the woman that if she ever needed to kill a hen again, just call me.
As a mink farmer I have had experience visiting abbatoirs to get food for my mink. There is a method of stunning before actually killing. However, there were/are sadists among the staff who have no compassion for the animals and if an animal became a problem for them they would cut off its ears or use their knives and saws on the creature. Such behaviour saddened me very much.