Born in Cociuba Mare, Bihor County, Șut played youth football for local side Izvorul, Liberty Salonta, and ASA Târgu Mureș.
[1] Șut went on to represent Unirea Jucu, Pandurii Târgu Jiu, and Academica Clinceni in the lower leagues,[2] before returning to the top flight by signing for FCSB in the summer of 2019.
[3] Șut made his debut for FCSB on 24 September 2020, starting in a 0–2 loss to Slovan Liberec in the UEFA Europa League third qualifying round.
[5] On 22 May 2022, Șut scored twice in a 3–1 win over CFR Cluj, which had mathematically clinched the Liga I trophy the previous fixture.
On 7 June 2024, Șut was selected in Romania's squad for the UEFA Euro 2024,[6] but did not feature in any match.