You are applying ‘begotten’ in a way it was not intended. Would you make God an amoeba, reproducing himself?
The begetting of the Son occured in time with the birth of Christ.(Psa.2, Lk.1)
That is why the KJB correctly has in John 1:14,
Joh 1:18 No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.
The NASB has begotten God, and God cannot be begotten as God. You end up with a lesser ‘god’.
What He begot for the 2nd Person of the Trinity was the perfect human qualities,(Heb.10:5) (combined with the pure human nature of Mary, the woman doesn't transmit the Old Sin Nature, the man does) so that the Word became the God-Man, without Adam's sin and having two natures (God and Man) and one person.