It doesn't seem like there is a practical way for a person to euthanize pinky mice.
a. Mouse, Rat, and Hamster Neonates up to 10 days of age: Acceptable methods for euthanasia include: injection of chemical anesthetics (e.g., pentobarbital), decapitation, or cervical dislocation. Additionally, these animals are sensitive to inhalant anesthetics; e.g., CO2, or isoflurane from a vaporizer (used with appropriate safety considerations) although prolonged exposure, up to 50 minutes1, may be necessary. A secondary physical method of euthanasia is recommended to ensure death (e.g. cervical dislocation, decapitation, bilateral pneumothorax). “Fetuses that are believed to be unconscious and altricial neonates < 5 days of age ... may be quickly killed by rapidly freezing in liquid N2.”1 For neonates 5 days or greater, immersion in liquid nitrogen may be used only if preceded by anesthesia. Anesthesia may be induced by inhalant or injectable anesthetics; the institute veterinarian should be consulted for appropriate agents and dosages. Alternatively, when adequately justified, hypothermia may be used to induce anesthesia in pups six days of age or less (however 3-4 days of age is more typical).