[2][3] Madden was selected in the 34th round of the 2018 MLB Draft by the Kansas City Royals but opted not to sign with the team.
[6] As a sophomore, he served as the Longhorns' Saturday starter and went 3–0 with a 1.80 ERA in four starts before the season was cut short due to the coronavirus pandemic.
[7][8] Madden entered his redshirt sophomore season on the watchlist for the Golden Spikes Award and a preseason All-American by multiple outlets, as well as a top pitching prospect in the upcoming draft.
[9][10] Madden was named first team All-Big 12 and the Conference Pitcher of the Year after posting a 7-5 record with a 2.45 ERA and 137 strikeouts in 113+2⁄3 innings over 18 starts.
[16] He made his major league debut that day against the Chicago White Sox, tossing 5 innings and allowing 2 runs (1 earned) on 4 hits with 2 strikeouts.