2010-11: Brandon Halverson played for the Victory Honda bantam AAA team in Michigan and was invited to USA Hockey's Warren Strelow National Goaltending Camp in May 2011.
2011-12: Halverson played for the Little Caesars U16 midget team in Michigan. In 20 games for Little Caesars he was 6-10-2 with 1 shutout and had a 3.09 goals against and .897 save percentage. Halverson was selected by Sault Ste. Marie in the seventh round (127th overall) in the 2012 OHL Priority Draft.
2012-13: Halverson attended OHL camp with the Greyhounds before joining the Oakland Junior Grizzlies U18 team in Troy, Michigan. Splitting the goaltending duties with Michael Tilford, he was 9-9-2 in 21 games and had a 2.88 goals against and .906 save percentage. Halverson started all four playoff games and was 2-1-1 with a 1.46 goals against and .953 save percentage. He participated in USA Hockey's Select 17 Camp in Rochester.
2013-14: Halverson joined the Sault Ste. Marie Greyhounds as a backup to 19-year-old Penguins' prospect Matt Murray in his first OHL season. Halverson was ranked ninth amongst North American goaltenders in Central Scouting's mid-term rankings prior to the 2014 NHL Draft.