[4] Burcham played college baseball for the Sacramento State Hornets, of the Western Athletic Conference (WAC).
[16] In 2015, he played for the Boise Hawks of the Class A (Short) Northwest League, posting a .235 batting average with two home runs and 11 RBIs.
Burcham played in 2016 for the Asheville Tourists of the Class A (Full) South Atlantic League, batting .271/.338/.382.
[18][19] In the 2018 season Burcham played for the Hartford Yard Goats of the Class AA Eastern League.
[20] He missed nearly a month of action during the season in order to play for Team Israel at the 2020 Summer Olympics.
[23] On February 25, 2022, Burcham signed with the Southern Maryland Blue Crabs of the Atlantic League of Professional Baseball.
[13] On April 28, 2022, Burcham was traded to the Charleston Dirty Birds of the Atlantic League in exchange for a player to be named later.
[29] He was the starting shortstop for Team Israel at the 2017 World Baseball Classic in the main tournament, in March 2017.
[30] In the tournament's opener, Burcham singled in the game-winning run in the 10th inning in a 2-1 win over South Korea.
[18] In November 2019, he obtained Israeli citizenship so that he could play for Team Israel at the 2020 Summer Olympics in Tokyo.