In 2021, Bigge made 17 appearances for the High–A South Bend Cubs, logging a 5.66 ERA with 29 strikeouts over 20+2⁄3 innings pitched. He missed playing time during the season as the result of a torn labrum.[2][5] Bigge split the 2022 campaign between South Bend and the Double–A Tennessee Smokies, accumulating a 7–2 record and 5.26 ERA with 70 strikeouts and 10 saves in 41 games.[6] In 2023, he played for Tennessee and the Triple–A Iowa Cubs, compiling a 4.50 ERA with 68 strikeouts across 42 appearances out of the bullpen.[7]
Bigge began the 2024 campaign with Triple–A Iowa. On July 7, 2024, the Cubs selected Bigge to the 40-man roster and promoted him to the major leagues for the first time.[5] He made his MLB debut on July 9,[8] and recorded a 2.70 ERA with 5 strikeouts in 4 games.