According to trbo20 :
Remove the outlet

Widen the hole so it exposes half of the left stud and cut the top of
the hole back to expose a few more inches above the outlet box.

If the stud on the right isn't doubled, and I was trying for best
possible job, I'd widen that side too, so I could put a batten
behind it to support at least the right side of the hole.

[The idea being making sure that the right side of the hole
can't push back and crack the tape. That may be overly fussy
for many. Depends on how much of a span there is on the right.
Etc]

All screwed down, no glue required.

Further, as the outlet ears are supported at the top of the hole by
the existing drywall, it isn't necessary to extend the hole _up_ any
further, because the outlet cover will reach far enough "up". The patch
can be "U" shaped. Makes cutting/layout of the patch easier.

Cut a square patch that fits as accurately as you can.

Put a little chalk on the box and push the patch into the hole
transferring a chalk outline of the box opening onto the back of it.

Using a small keyhole saw or utility knife, very carefully cut the box
opening in the patch.

Glue the patch to the studs, tape, and mud the seams. Sand, prime, and
paint

Put the outlet back on.

As for the vanity splash, I would attempt to have the patch dip under
it by a few inches and glue it in place. Use caulk to hide any
imperfections.

You may not be able to get a couple inches behind the vanity and at
the same time be able to install it over the electrical box. The
studs and backsplash form a "slot" that the drywall MUST follow,
but the outlet box is protruding in the "plane" of the slot. So
something has to bend (unless there's enough slop in the slot).
Regular drywall doesn't bend very well.

Even 1/4" of overlap would be enough, but 1/2" would be better in
terms of ensuring that the edge doesn't crumble away, and still be
possible to slide it in. [As it is, you'd probably have to bevel
the back side of the patch on the bottom.]

If you really had to have several inches below the backsplash,
you'd have to remove the outlet box, install the patch, and then
reinstall a retrofit box.

But if you're good with priming and caulk, you could probably get
away with no overlap at all.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.