Adding a receptacle to a wall with an insulation blanket on the wall
Anything less than 1-1/2 plus the sheetrock is about the shallowest you can do and meet code.
This will hold a receptacle and allow a nm cable routed down the wall but no room for splices in the receptacle box.
the 1-1/4” space then the nmb on the wall but trying to put this in after the fact it would be best to surface mount. As junction boxes at the ceiling would need to be accessible or each run a home run lots of wire.
Surface mount conduit can be done it may not look as nice as hidden but is more common in basement spaces and may be the only legal way depending on how thick the insulation board is.