College career
Broome was named Northeast Conference Player of the Week three times as a sophomore. On February 20, 2016, he became the only player since the 2010–11 season with least 39 points, six rebounds, and six assists without turning the ball over. Broome was named Northeast Conference Men's Basketball Player of the Year in his sophomore season at Sacred Heart, just the second sophomore to do so. On the season, he averaged 23.1 points and 4.9 rebounds per game.
Broome announced in April 2016 that he would transfer to Cincinnati and sit out a season. It was an adjustment, as Broome was used to being the star but had to become a role player. As a junior, he scored in double figures seven times. Broome averaged 7.9 points per game as a junior on a 31–5 Cincinnati team that earned a 2 seed in the NCAA Tournament. As a senior, he averaged 8.3 points and 2.2 assists per game. He considers his 16-point, five-assist performance in a loss to Xavier on December 2 to be his best game.
Professional career
Broome signed with the St. John's Edge of the National Basketball League of Canada on October 11, 2019. On February 18, 2020, Broome was named National Basketball League of Canada player of the week after averaging 22.5 points per game in two victories. He averaged 13.9 points and 2.1 rebounds per game, earning league All-Rookie Team honors.
After the Edge folded in 2021, Broome became an assistant basketball coach at his high school alma mater, St. Thomas More School in Oakdale, Connecticut.
Redshirt