how to transfer ownership of private illiquid shares and what is cost basis?

I own a few shares of a private firm. The shares are registered at Carta, but trading is not allowed.

How can I transfer the ownership to my son so that the cost basis would be the price I bought them for originally?

Suppose I bout the shares when they were issued at the nominal price of $100. There is no market for these shares, and I can’t transfer rights to my son on Carta itself, i.e. register them on him. However, I want to somehow transfer the ownership to him in such a way that from tax point of view it will be a sale at $100. So that my capital gain is zero. He gets the ownership at $100 cost basis.

Since there is no market, there is no market price for shares. At earlier financing rounds they were valued at $400, but there was no way to actually sell them at this price. After this year market events, it’s not clear what’s the value anyways. The accountants told me that there was "tax event" for my ownership so far, i.e. they must be still valued at $100 from tax point of view.