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. In 19 games for the Greyhounds he was 12-6-1 with 2 shutouts and had a 2.96 goals against and .904 save percentage. Sault Ste. Marie finished first in the West Division and reached the second round in the playoffs. Murray started all nine playoff games for the Greyhounds. Halverson was ranked sixth amongst North American goaltenders in the Central Scouting final rankings and was selected by the New York Rangers in the second round (59th overall) in the 2014 NHL Draft.
2014-15: Halverson played in 50 of 68 regular season games for Sault Ste. Marie in his second OHL season and played for the USA U20 team in the 2015 World Junior Championship. He was 40-5-2 with 6 shutouts and had a 2.63 goals against and .913 save percentage for the Greyhounds. Sault Ste. Marie finished first in the West Division and reached the Western Conference finals, falling to the Erie Otters. Halverson was 10-4 with 1 shout and had a 2.94 goals against and .893 save percentage in 14 playoff games. Backing up Vancouver prospect Thatcher Demko for Team USA, he recorded a shutout in his only start, stopping 14 shots in a 6-0 win over Switzerland. The USA finished fifth, falling 3-2 to Russia in the quarterfinals. Halverson signed a three-year entry-level contract with the Rangers in July 2015.