The English translation is nonsense. MAYBE you could say "the man's quick career continued when he became the boss" (not "as" he became the boss), but it's still not something we would say in English.
Why not "as"?
Quick is a terrible, terrible adjective here - swift is the most likely synonym to describe a career.
Why 'continuing'? Hoovering over 'fortsættes' only gives 'continued' and to be frank, 'was continued' sounds much better too, in a past passive lesson.
Also, "...he became a boss.", yes?
What makes it mean "was continuing" as opposed to "was continued"?
Honestly, the pedantry required to get this right is huge. 'chef', here, means 'the boss', not 'boss'.
It should be boss not "the boss"