What is the best, the most humane way, to euthanize a bird, mouse or baby bunny, e.g., that has been critically injured, beyond hope, by your dog or cat?

How do you do it, if you do?

I thought I was done with this heartbreak when I got dog, after a lifetime of cats, but my dog got hold of field mouse last night… the first time in the almost two years I have had her that she has done anything like that.