Døssing was born in Viborg, Jutland and started his career with his hometown club Viborg FF in 1958, aged 17. With the help of Jack Johnson, a former manager of B 1913, Døssing got in contact with Dundee United.
Døssing's professional football career began at Viborg FF in 1958, while he was still a teenager. He was helped by Jack Johnson, a previous manager of B 1913, to connect with Dundee United.