Pet starts to sleep, soundly ensconced in bed (6)
I believe the answer is:
cosset
'pet' is the definition.
(thesaurus)
'starts to sleep soundly ensconced in bed' is the wordplay.
'starts to' says to take the initial letters.
'in' means one lot of letters goes inside another.
'bed' becomes 'cot' (cot is a kind of bed).
The first letters of 'sleep soundly ensconced' is 'sse'.
'sse' put within 'cot' is 'COSSET'.
(Other definitions for cosset that I've seen before include "Pamper, mollycoddle" , "Pamper and spoil" , "Protect overindulgently" , "Fondle" , "lamb" .)