Best method up to code to project an existing electrical outlet to a interior media built in cabinet face 3 1/2" farther out
You'll have to move the electrical box

You can't entrap an electrical junction box behind built-in furniture like that.  Every box cover must be accessible without tools (obviously you need a screwdriver to remove the box cover itself, but those are the only screws you are allowed to turn). 

Further, you cannot splice anywhere but a junction box, and again, junction boxes must be accessible.  

So lining up the framing of the cabinet with the electrical box is not really an option. The framing must move, or the box must move. 

If you have a flat space on your built-in cabinet that aligns with the existing box, you could just stack box extensions to get where you need.