Is it grammatically possible to begin a sentence with "but" ?
According to the Cambridge Dictionary, the following case (but it is not the only valid case) may apply
But for + reason
But for is used to introduce the reason why something didn’t happen:
But for the traffic, I would have been here an hour ago. (The traffic was very heavy – if it weren’t for the traffic, I’d have been here an hour ago.)
They would have been badly injured but for the fact that they were wearing seat belts. (They were wearing seat belts – if it weren’t for the fact that they were wearing seat belts, they would have been badly injured.)