To be honest if a person wants a basic yet decnet driver circuit for any type of inverter Would recomend going out and buying a actual dedicated IC that does the job right.
There are hundreds to choose from and everyone of them will do the job better and with less parts than any toss together circuit made from 555's and logic gate bits N pieces.
I would recomend starting with something like a LM494 or a LM3524 IC or one of their many variants as a basic driver IC and go from there.