Brossoit played four seasons (2009-13) for Edmonton of the Western Hockey League. In 2012, he went 42-13-5 with a 2.47 goals-against average and a .914 save percentage, leading his team to the WHL championship and a berth in the Memorial Cup in 2012.
Selected by the Calgary Flames in the sixth round (No. 164) of the 2011 NHL Draft, Brossoit signed a three-year entry level contract on April 4, 2013. He was acquired by the Edmonton Oilers in a trade with the Flames on Nov. 8, 2013.
Brossoit made his NHL debut April 9, 2015, making 49 saves (two goals allowed) in a 3-1 loss against the San Jose Sharks. He went 0-4-1 with a 3.61 GAA and .873 save percentage in 2015-16. On Jan. 21, 2017, Brossoit made 38 saves to earn his first win, 7-3, against the Flames. He finished the 2016-17 season 4-1-0 with a 1.99 GAA and .928 save percentage.
After going 3-7-1 as a backup for the Oilers in 2017-18, he signed with the Winnipeg Jets on July 2, 2018. Brossoit flourished in his first season with his new team, going 13-6-2 with a 2.52 GAA, a .925 save percentage and one shutout in 21 games (19 starts).
Brossoit signed a one-year, $1.75 million contract to return to the Jets on July 1, 2023. He was 7-0-3 with a 2.17 GAA and .927 save percentage in 11 regular-season games (10 starts) for the Vegas Golden Knights in 2022-23, and 5-2 with a 3.18 GAA and .894 save percentage in the Stanley Cup Playoffs. He did not play after sustaining a lower-body injury in Game 3 of the Western Conference Second Round against the Oilers.
Brossoit signed a two-year, $6.6 million contract with the Chicago Blackhawks on July 1, 2024.
