2007-08: Brady Brassart played bantam hockey in suburban Vancouver. He was selected by Spokane in the second round (43rd overall) of the 2008 WHL Bantam Draft.
2008-09: Brassart played for the Vancouver NW Giants midget major team in British Columbia. He scored 24 goals with 31 assists and 60 penalty minutes in 40 regular season games. Brassart scored 4 goals with 3 assists and 2 penalty minutes in five playoff games.
2009-10: Brassart skated in 60 games, including seven playoff contests, with the Spokane Chiefs in his first WHL season and played for Canada Pacific in the 2010 U17 World Hockey Challenge. He scored 9 goals with 6 assists and was +6 with 15 penalty minutes during the WHL regular season. Spokane finished third in the U.S. Division. Brassart had 1 assist and was -4 with 4 penalty minutes in the Chiefs' seven-game playoff series with Portland. In five games for Canada Pacific he scored 1 goal with 2 assists.
2010-11: Brassart played in 65 of 72 regular season games for Spokane in his second WHL season. He scored 8 goals with 24 assists and was +14 with 70 penalty minutes. The Chiefs finished second to Portland in the U.S. Division and lost to the Winterhawks in the Western Conference finals. Brassart scored 1 goal with 2 assists and was +5 with 10 penalty minutes in nine playoff games.
2011-12: Obtained by the Calgary Hitmen in a WHL trade with Spokane in August 2011, Brassart was the team's fourth-leading scorer while finishing second amongst Hitmen forwards with 106 penalty minutes. He scored 25 goals with 34 assists and was +21 in 70 regular season games. Calgary finished second to Edmonton in the Central Division. Brassart was -3 with no points and 8 penalty minutes in the Hitmen's five-game playoff series with Brandon. Brassart was ranked 181st amongst North American skaters in Central Scouting's final rankings prior to the 2012 NHL Draft but was not selected.
2012-13: Brassart was the third-leading scorer for the Calgary Hitmen in his fourth WHL season. He scored 35 goals with 43 assists (both career-highs) and was +40 with 88 penalty minutes in 65 regular season games. Calgary finished second to Edmonton in the Central Division and lost to the Oil Kings in the Eastern Conference finals. Brassart scored 9 goals with 1 assist and was +1 with 22 penalty minutes in 17 playoff games.
2013-14: Brassart went to training camp with Tampa Bay but did not sign a contract with the Lightning before returning to Calgary for his 20-year-old season. Brassart shared the scoring lead for the Hitmen with 18-year-old Greg Chase as both finished with 35 goals and 50 assists in 70 regular season games. Brassart, an assistant captain, finished +11 with 94 penalty minutes. The Oil Kings and Calgary each finished with 103 points; with the Hitmen finishing first in the Central Division based on more wins. Brassart scored 3 goals with 6 assists and was -1 with 8 penalty minutes as Kootenay defeated Calgary in a six-game first round playoff series. He signed a three-year, entry-level contract with Minnesota as a free agent on March 1st, 2014.
The Wild signed Brassart to a three-year entry-level contract on March 1st, 2014.