The action I'm repeating over and over is transplanting my changes from default
to production branch. However, each time this means a tedious and error prone task of hand picking the changesets I want to transplant between plethora of other changesets.
Is there any way of filtering available chagnesets by author and message?
From the screen shot, no. It looks like you can't. As a commenter mentioned, you can use the command line or TortoiseHG to query using a revset query:
hg log -r "user('vartec') and desc('needle')"
In Tortoise HG, click the "Search" icon (the magnifying glass) and enter the revset query in the search box.