Cole also earned a five-minute major for kneeing on the play, which took place with just over six minutes to play in the second period.
Shore was helped off the ice by teammates and headed straight to the Ducks' locker room. He did not return.
Cole avoided supplementary discipline for a similar knee-on-knee collision with New York Islanders forward Brock Nelson earlier this season. He was assessed a two-minute minor on that play.
The 30-year-old rearguard earned a three-game ban in 2012 for an illegal check to the head of Detroit Red Wings forward Justin Abdelkader, the only suspension of his career.