After breaking and replacing several panes of mica in our antique parlor stove trying to keep them clean enough to watch the fire, we found that they clean amazingly easy if we spray the panes with a light coat of clear silicon spray after we clean them.
Now we just wipe or spray a little degreaser on them (Simple Green is great) and rinse the doors under the faucet or wipe them VERY gently. The soot seems to fall off!
Our antique Amherst burner has three sides with mica glass doors, 22 panes on the front and 15 panes on each side. A real pain to clean at first!! But oh, so beautiful to watch the logs or coals burning and warming our home!
Now we just start the morning with a quick rinse and wipe of each door, then re-spray of the clear silicon before we light it for the day.
Hope this helps