During that time, he played football in the academy of local Danish Superliga club FC Midtjylland, but when the school year ended, he returned to his hometown, Bogense, while his career continued in the youth of OB.
[2] Midtjylland's reserve team, Ikast fS experienced some issues during their 2008–09 campaign, and in the summer of 2009, Bøge and two other teammates were promoted to the squad.
[3] However, he made his debut for Midtjylland on 6 December 2009 against FC Nordsjælland, where he replaced Jonas Borring in the 76th minute; his only appearance during the season.
Bøge did not make an appearance during the season, and afterwards stated that he was interested in being loaned out, due to a lack of playing time.
Bøge quickly became an important player for the team and extended his contract for two years, after a successful 2013–14 season with 32 league appearances, as Hobro won promotion to the Danish Superliga for the first time in club history.