[2] Kaše spent the entirety of the 2015–16 season with the Ducks' American Hockey League (AHL) affiliate, the San Diego Gulls.

On 4 November 2016, Kaše recorded his first career NHL point, assisting on a goal by Antoine Vermette during a 5–1 win over the Arizona Coyotes.

Once the season resumed in the playoff bubble, Kaše found some offensive success, with four assists in 11 games before the Bruins were eliminated by the Tampa Bay Lightning in the second round.

[8] On 25 April 2022, Kaše was chosen as the Maple Leafs' nominee for the Bill Masterton Memorial Trophy for his dedication and perseverance to the sport after a series of serious head injuries.

[13] Serving as an assistant captain, Kaše had a successful season with Litvínov, scoring at above a point-per-game pace, with 54 points in 48 games, including 23 goals.

Kaše would help lead Litvínov to the semi-finals of 2023–24 Czech Extraliga Playoffs, where they would be swept by the top seeded HC Dynamo Pardubice.

Kaše's younger brother, David, plays alongside him for HC Litvinov and was selected in the fifth round, 128th overall, by the Philadelphia Flyers in the 2015 NHL Entry Draft.