Brossoit was selected by the Calgary Flames in the sixth round, 164th overall, of the 2011 NHL Entry Draft and played for the Edmonton Oilers and Winnipeg Jets prior to joining the Golden Knights.

Brossoit played for Team Pacific at the 2010 World U-17 Hockey Challenge, posting a 2.00 GAA and a .902 save percentage in two games.

[6] On April 2, 2013, Brossoit was named as Canadian Hockey League (CHL) Goaltender of the Week after posting a 4–1–0–0 record with two shutouts along with a 0.76 GAA and .968 save percentage in the opening round of the WHL Eastern Conference playoff series that saw the Oil Kings defeat the Kootenay Ice four games to one.

[16] On July 28, 2021, following his third season with the Jets, Brossoit left as a free agent and was signed a two-year, $4.65 million contract with the Vegas Golden Knights.

[21][22] Injuries to both Logan Thompson and Adin Hill eventually prompted Brossoit's recall to the NHL, playing alongside former Los Angeles Kings starter Jonathan Quick, who the team acquired midseason.

[23][24] The Golden Knights finished first in the Western Conference to qualify for the 2023 Stanley Cup playoffs, with Brossoit named the starting goaltender for their first-round series against the Winnipeg Jets;[25] Brossoit's starting role against his former team and partner Connor Hellebuyck attracted media attention.

[29] The Golden Knights defeated the Florida Panthers in five games, and Brossoit joined the team in hoisting the Stanley Cup.

[31] The 2023–24 season proved a great success for the Jets, who finished fourth in the league, while Brossoit called it "a blessing to be here" while seeking to reestablish his ability to play and remain healthy after previous injuries.

He was the Jets' nominee for the Bill Masterton Memorial Trophy, awarded by the Professional Hockey Writers' Association to the player who "exemplifies the qualities of perseverance, sportsmanship and dedication to the game.

[33] As a free agent at the conclusion of his one-year deal with the Jets, Brossoit was signed to a lucrative two-year, $6.6 million contract with the Chicago Blackhawks on July 1, 2024.