Do boutique investment banks advise on their own IPOs?

Sorry to be that guy, but neither of the transactions you mentioned were actually IPOs. Shocker. PWP merged with FinTech Acquisition Corp IV, a SPAC, in order to go public, and was the exclusive capital markets advisor to itself in doing so, so what you're saying is possible. The reverse merger with the SPAC happened in 2021, but in 2018, PWP had actually hired Goldman Sachs and JP Morgan regarding their options for an IPO. That fizzled, and then they pulled the trigger on the de-SPAC transaction.

PJT was a separate situation. PJT was a unit of Blackstone, which was publicly traded at the time, and Blackstone spun off the PJT unit into a standalone company, with BX shareholders receiving 65% of SpinCo and the titular Paul J. Taubman and his people holding the rest. BX had no financial advisors on this transaction, and had Simpson Thacher as legal counsel. They basically needed to negotiate the legal agreement with Paul Taubman and then the rest was very plain vanilla: give shares of SpinCo pro-rata to the existing BX shareholders. Nothing too complicated.

Goldman Sachs, Lehman Brothers, and UBS underwrote Greenhill. 

BAML, Goldman Sachs, and a big syndicate underwrote Houlihan Lokey.

Lehman Brothers underwrote Evercore.

Goldman Sachs underwrote Lazard.

It all depends, but it seems that most of the time, investment banks are not their own investment bankers on their IPO. This happens for at least 2 reasons. 1) The amount of money needed to be raised is often too much for one bank to handle on its own, so it brings in buddies to help. 2) Some of the shareholders of the bank going IPO may not work at the bank, so it's a conflict of interest for management to handle everything on their own when there are outside shareholders too. You can solve that by bringing in a 3rd-party.