Final answer:
Ellison 'Cotton Top' Mounts was the only Hatfield member to be executed for his role in the New Year's Day shootout in 1888, reflecting the often violent and lawless nature of frontier justice.
Step-by-step explanation:
The only Hatfield member to receive capital punishment for the shootout on New Year's Day, 1888, was Ellison 'Cotton Top' Mounts. He was implicated in the infamous Hatfield-McCoy feud, a violent and lengthy confrontation between two families that epitomized the concept of family feuds in the United States' history, particularly in the Appalachian region.
Although the feud involved many skirmishes and numerous deaths over several years, Mounts was the only member of the Hatfield family who was eventually executed as a result of the conflict.
This episode of American history reflects the broader context of frontier justice, where violence was common and legal proceedings were often informal or influenced by local power dynamics.