Can I recess electrical panel deeper than usual?
Yeah, no.  Panels do not work like switches and outlets.  You can't jack the panel cover outward and use longer screws and have a drywall trim ring between panel cover and panel.  No way.  Not least, the breakers will fall out.  Because the relationship between cover and panel must be precise; it's the only thing holding them in!  



You also can't bury the panel cover behind drywall, because as any regular visitor here has memorized, panel covers must be accessible without any tools and without modifying the house in any way. 

Your only option is to stop the additional drywall layer a couple inches before it reaches the panel cover, and hide the jagged drywall edge with a simple piece of wood trim around the perimeter, like a picture frame.  Finally, a use for that miter box! 



ThreePhaseEel makes the point that a "picture frame" (and the drywall itself) intrudes on the "box" of open space that must be kept clear for a person to access and maintain the panel. That open-space "box" starts at the face of the panel and is 36" deep. It is 30" wide but need not be centered, and is 78" (6'6") tall. 

However, I see no problem with what is described, for obvious reasons.  I don't stand with my toes planar with the front of the panel (that would be very cramped, eh?) so I don't care about the wall being 3/4" thicker. Disclaimer here: I have an installation just like OP, that violates this in a statutory way, but is of no genuine inconvenience to the installer, and I really like the installation (it's a tasteful cabinet door concealing electrical and phone boxes). So of course I'd say this, wouldn't I? 

However you would either need to get a variance from your AHJ, which I would think would be no trouble; or simply have the "picture frame" stop short of the open-space "box".  In my case, the cover was done by a predecessor whose other work is impeccable, so I presume he got a variance :)