How do I secure a TV wall mount?
Now that I see your photo I'd do this differently. I'd install a vertical cleat just behind the faceframe on each side of the cabinet, maybe 1" back (the thickness of the plate plus 1/4"). I'd then span a sheet of 3/4" plywood across them, creating a solid face on which to install your mount. 

You should have either a cabinet wall or framing to screw into. Make the plywood as tall as possible without being visible to give it rigidity. You could also use two-by lumber if you prefer.

Plan view:

    | |                                      | |
    | |_                                    _| |
    | | | &lt;-- cleats --------------------&gt; | | |
    | |_|__________________________________|_| | &lt;-- cabinet wall
    | |______________________________________| |
    |_|     ^ -- plywood plate               |_|
  ________________________________________________
 /                  TV                            \
|__________________________________________________|


One advantage to this approach is that you can use a simple, inexpensive mount rather than an elaborate articulating arm mount. This saves money and results in a more stable setup. Be sure to work out the depth of everything so you're able to tilt as you like when you're done.



Original answer:

Rather than installing something that may be visible around the television, just add a sheet of 3/4" plywood to the one stud, and also adhere it to the drywall with project adhesive or silicone. 

Cut a sheet that's about 6" smaller than the tv in both directions. Apply a 1/4" bead of adhesive to the entire back side in a zigzag pattern. Using four 2-1/2" gold screws, fasten the sheet to the stud Once the adhesive is set this will hold any modern tv with no problem. 

Be sure to properly pilot your mount screws so they grab the plywood well. If you're using an articulating mount, consider installing a pair of toggle bolts at the ends of the plywood as insurance in preventing the sheet from pulling away.

Elevation view:

------------------------------------------------------
.                                                    .
.     __________________________________________     .
.    |                    *                     |    . &lt;-- TV outline
.    |   (       (                  (      (    |    .
.    |    )       )       *          )      )   | &lt;- . --- plywood
.    |   (       (                  (      (    |    .
.    |    )       )       *          )      ) &lt;-| -- . --- adhesive
.    |   (       (                  (      (    |    .
.    |    )       )       *          )      )   |    .
.    |__________________________________________|    .
.                                                    .
.                         ^--- screws                .
------------------------------------------------------