I would say neither of these sentences is correct English grammar. The first one might be correct if the two sons somehow became a single doctor. The second one is just incoherent. Also, you need "became", since this is in the past.
I think you're trying to draw a distinction between:
He has two sons who both became doctors.
He has two sons; both became doctors.
In this case, the first one leaves open the possibility that he has more sons. The second does not, since it makes "He has two sons" it's own thought. Strictly speaking, you could say "He has two sons" to mean he has at least two sons, but nobody does that.
(Like the old puzzle, "A man has two ordinary American coins that total 35 cents. One of them is not a quarter. What are they?" The gag answer: "The one that's not a quarter is a dime, the other one is a quarter.")