Seeing American lawyer can start to grind (6)
I believe the answer is:
dating
'seeing' is the definition.
('date' can be a synonym of 'see')
'american lawyer can start to grind' is the wordplay.
'american lawyer' becomes 'da' (abbreviation for District Attorney).
'can' becomes 'tin' ('tin' can be a synonym of 'can').
'start to' indicates taking the first letters.
The initial letter of 'grind' is 'g'.
'da'+'tin'+'g'='DATING'
(Other definitions for dating that I've seen before include "Taking out" , "Going steady" , "Imposing time code or keeping social engagement" , "Seeing" , "Going out with (a person)" .)