I am not a teacher, but I am a native speaker.
In English, when someone says "he eats his vegetables", which could also be "he eats his carrots" I suppose, it is meaning to say "he is strong" or something almost like that.
It seems like Carmela is saying the phrase out of nowhere, but when I thought about it, she may be saying it about Tony to say that he is being very blunt and strong in the way he is saying his opinion to her about her hair.
That is my take on it at least.
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