Barré-Boulet played at the midget level in his native Quebec with both the Rive-Sud Express Midget Espoir and Lévis Commandeurs of the Quebec Junior AAA Hockey League (QMAAA) from 2012 to 2014. He was a late round draft pick to the Drummondville Voltigeurs (102nd overall) in the 2013 Quebec Major Junior Hockey League (QMJHL) Entry Draft. Developing early as an offensive under-sized forward, Barré-Boulet lead the Voltigeurs with 54 assists, placing second in team scoring with 89 points in just his second season in the QMJHL in 2015–16.
During his third season in the QMJHL in 2016–17, Barré-Boulet was traded by the Voltigeurs to the Blainville-Boisbriand Armada in exchange for two players and three draft selections on January 6, 2017.[1] In scoring at a greater than point-per-game pace with the Armada, Barré-Boulet recorded 14 goals to lead the league in that year's playoffs.[2]
Having ended his junior career, Barré-Boulet attended both the Tampa Bay Lightning's rookie and main training camps, before he was reassigned to American Hockey League (AHL) affiliate the Syracuse Crunch to begin the 2018–19 season. Barré-Boulet scored the game-winning goal in his professional debut on October 6 and went on to be the top-scoring rookie in the AHL during the regular season with 68 points in 74 games for the Crunch, placing sixth in the league's overall scoring race. He earned a share of the Willie Marshall Award as the league co-leader with 34 goals, alongside teammate Carter Verhaeghe.[6] He likewise won the Dudley "Red" Garrett Memorial Award as most outstanding rookie.[7]
Barré-Boulet was one of the eight players called up to the Lightning for their training camp prior to the 2020 Stanley Cup playoffs.[10]
On February 22, 2021, Barré-Boulet skated in his first career NHL game in a 4–2 Lightning win over the Carolina Hurricanes at PNC Arena.[11][12] On April 25, he recorded his first career NHL goal and point in a 4–3 Lightning overtime win over the visiting Columbus Blue Jackets at Amalie Arena.[13] Barré-Boulet subsequently signed a three-year contract exentension to remain with the Lightning in July 2021.[14]
On October 10, 2021, he was placed on waivers by the Tampa Bay Lightning, and was claimed by the Seattle Kraken the next day.[15] He played two games for Seattle recording an assist before, on October 21, he was placed on waivers by the Kraken, and was reclaimed by the Lightning the next day.[16]
Barré-Boulet's wife, Anne-Marie, gave birth to their first son, Zack, on February 16, 2022.[19] The couple welcomed their second child, a daughter named Anna, in March 2023.[20]