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.
I'm surprised this request hasn't gotten more traction yet. Worked late into the night to get some things done but work was still unfinished. Came back to my desk this morning and of course, since the MacBook had slept, the connections had timed out as well. Tried reconnecting, and lost all history. Now I'm not sure what all I had achieved last night because of late-nighter brain-fog.
as mentioned above, PuTTY (the executable we start) is not from us and we can't change the behavior. If the process dies, the only thing we can do is to restart it. If session data is lost in the process, there's nothing we can do on our side. As others have mentioned, when PuTTY loses the connection, it stays open and you can use the "System Menu" in the Actions tab to reconnect the session WITHOUT restarting PuTTY. This should then also keep your history.