[Need advice] Company wants me to resign from my current job but refuses to give me a formal offer letter

I currently work at a startup (call it Company A) and I have a 3 month notice period (negotiable).

I was interviewing at another startup company (call it Company B) this week and they verbally offered me the job, but they said they wanted me to join ASAP (in just a week or two). That same day, I asked for an official offer letter to be sent to me, and they agreed.

The next day, Company B sends me an email (not a document or anything, just a plain email), stating my monthly pay and my designation. I ask again for the official offer letter, but the guy says he cannot issue it unless I speak to my current boss about when I can be relieved and communicate to him the joining date.

This would entail that I tell my boss at Company A about the verbal job offer (which would also reveal to him that I've been job-hunting) - without actually having a formal, signed job offer in my hand.

What's more, Company B tells me - "Don't tell your boss you're thinking of leaving. Just tell him decisively that you're leaving and work on your relieving date" - all before they've even given me an offer letter.

Is this how it normally goes while changing employers? From my understanding, one must have a signed offer letter before putting in their papers. Company B's antics seem very unusual to me.

Can someone advise me on how to proceed with this?

Update - 8:30 PM - I called the founder of Company B and he finally agreed to send me the offer letter. But he insisted I turn in my resignation before that. Also, he hinted that if I wished to abscond from Company A, he is willing to provide me with a fake experience certificate from some other companies he knows.

This guy really has some integrity!