Barbecue bill's not completely on stream (4,5)

'barbecue' is the definition.
I know nothing about this answer so I cannot tell whether it can be defined by this definition.

'bill's not completely on stream' is the wordplay.
'bill' becomes 'charge' (synonyms).
'not completely' means to remove the last letter (the word is incomplete).
'on stream' becomes 'rill' (rill is a kind of stream. I am not sure about the 'on' bit.).
'charge' with its final letter taken off is 'charg'.
'charg'+'rill'='CHAR GRILL'

Can you help me to learn more?