Starts off puzzle solving slowly, teaser that demands attention (4)
I believe the answer is:
psst
'that demands attention' is the definition.
'starts off puzzle solving slowly teaser' is the wordplay.
'starts off' suggests taking the first letters.
The initial letters of 'puzzle solving slowly teaser' is 'PSST'.
(Other definitions for psst that I've seen before include "Sibilant interjection to call attention" , "Noise used to attract someone's attention privately" , "Sibilant sound to attract attention furtively" , "Listen!" , "Pay attention" .)