What's the best way to embed 14-3 electrical wire into a stud.?
Answers:
use a router to mill out a channel for the wire to be recessed in. I think this is the idea you had but a router would be much faster than a chisel. Better yet, rout out a channel on the backside of the piece of drywall being careful to only go deep enough to go around the wire. Most drywall is 1/2" thick. The wire is probably about a 1/4" thick. This would leave 1/4" of drywall. Routing a channel in the drywall will make it very weak so be careful when moving it. Also, don't nail into the stud where the wire is.
Another idea would be to add a thin strip to wood to every stud except the one with the wire on it. This would keep the wall flat and allow space for the wire.
Other answers:
get a really long drill bit and drill a hole through it
get a really long drill bit and drill a hole through it
Try furring strips if you can't move the wire. I believe the electrical code requires the wiring to be recessed at least 1", depending upon where you are.
You really need to move the wire. I know it is extra work, but if you do the simplest path and just channel out the drywall so it fits you will be leaving the wire very exposed. Anybody could come along after you and try to hang something there. If they drive a nail through that you have left a serious liabilty. The hole through the stud does not have to be straight, just drill it at an angle.
the best way is to drill a hole in the stud but according to the NEC (National Electric Code) the hole needs to be at least 1-1/4" away from the edge of the stud. If the hole is not 1-1/4" away from the edge OR if you chisel ("notch") it in then you will need to use a "nail plate". The nail plate has to be made of metal (strong enough to withstand a dry wall screw) and 3/16" thick. Secure the nail plate over the notched part covering the wire.