A tonsillectomy was a medical procedure in which a patient's tonsils were surgically removed.
In 2256, Hugh Culber said that he needed to assist the chief medical officer of USS Discovery with an Andorian tonsillectomy. (DIS: "Choose Your Pain")
Leonard McCoy had undergone a tonsillectomy sometime prior to 2267, but his tonsils eventually grew back as a side effect of the Omicron spores. (TOS: "This Side of Paradise")
Some used this term as a metaphor for flirtatious kissing. In 2378, while playing the EMH in the holographic novel Photons Be Free, B'Elanna Torres accused Marseilles of getting her out of sickbay just so he could give her patient a tonsillectomy. (VOY: "Author, Author")