Using a Dutchman to completely fill a mortise lock hole

I came across a This Old House video that demonstrates how to fill a mortise lock with a Dutchman. The video patches one side of the door.

I want to patch both sides so I can drill it out and replace it with a modern door lever.

In the video, I believe they are repairing a previous patch so they can reinstall a mortise lock. Because I want to fill the cavity, should I simply cut each side deeper and at different sizes so the smaller insert uses the larger insert for glue backing?

What’s the preferred method for completely filling the mortise with new wood?