Are you sure...
To go from 24" to 36" means a 6" piece on either side. The glass area will look dwarfed by the wide pieces on either side...JMO.
Regardless, remove the glass before it gets broken. What's the thickness of what you have now? If you add a piece on either side, sand it down to level it out, the thickness will be somewhat lee than when you started.
A butt joint would be the easiest and then dowel it in several places. A splined joint would be my next choice, more difficult but doable. You would rout out a groove using an edge guide on the door and on the same side of the adjoining piece for a spline about 1/2" thick running the full length. This increases the glue area while adding some strength mechanically.
Matching the same wood, Mahogany, may be an issue in color and grain... I have a buddy who makes beautiful raised panel doors who I can call for his opinion as to how to do it. I'll see what he says.
The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)