Asher had Tommy John surgery when he was 14 years old. He attended McKeel Academy of Technology through his sophomore year of high school. He then transferred to Charlotte High School during his Junior year, but later went on to graduate from Lakeland Senior High School.

Asher was drafted by the San Francisco Giants in the 23rd round of the 2010 Major League Baseball draft out of Lakeland Senior High School in Lakeland, Florida. He had agreed to an $80,000 signing bonus with the Giants, however the team decided not to sign Asher after a bone spur was revealed in his physical. He attended Santa Fe College for a year before transferring to Polk Community College. He was drafted by the Texas Rangers in the fourth round of the 2012 Major League Baseball (MLB) draft. He made his professional debut for the Spokane Indians that year. He pitched in 20 games as a relief pitcher, finishing with a 3.09 earned run average (ERA), five saves and 50 strikeouts over 35 innings.

In 2013, Asher was a starting pitcher for the Myrtle Beach Pelicans. He appeared in 26 games with 25 starts and went 9–7 with a 2.90 ERA and 139 strikeouts over 133 ¹⁄₃ innings.

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