unfortunately this is not possible. The PuTTY integration is done by invoking an external program (putty.exe itself) and when a new connection is established a new process is started.
I'm sorry I have no better news for you.
There is a reconnect feature in putty. If you have an open session that times out or closes, and you don't have putty set to close the window automatically, you can click the menu, and select reconnect. It will retain your history and log you back into the session.
This is something that should be doable even if you are invoking external applications. SuperPutty does this, SecureCRT (which uses its own terminal application) also does this. Putty Connection Manager used to do this as well.
For the use case: As a Network Engineer we have 5-10 sessions open to work on a design. This is a workspace for us. We then start a long download of software to update one of our devices which takes 30-45 minutes. During this time we are doing other things. An hour goes by we come back to Royal TS to check progress and all the sessions are closed. We have to read logs to see if the transfer completed.
when PuTTY itself lost the connection, you can use the System Menu button in the Actions ribbon tab to initiate the reconnect in PuTTY.