Show a screen shot it will be easy if the drop shadow is hard edge.
If the drop shadow and the letter is not grouped group them then in the Layers Panel target the group by clicking the little circle to the right of the lGroup name.
I the appearance panel from the flyout select new stroke you may have to re stack it but that should work.
You have to drag the stroke below the contents and give it a slight offset effect I had a 1 point stroke and in order to see the whole stroke gave it a.5 pt offset like such.
You can make this a graphic style as well.
Edit you might want to expand the appearance once you have this or fatten the transparency so as not cause complications for the RIP. Not sure that is necessary but in case you encounter difficulties that would be something to look into.
Assuming you're talking about cutting this on a knife plotter, you're going to have to do more than just apply an effect. Making something merely look right on the screen is a far cry from properly preparing it for cutting.
You need the artwork to be composed of only paths that describe the paths of the knife; no crosscuts across areas of contiguous color or self-crossing paths.
1. Select the artwork and Object>Paths>OutlineStrokes. That will convert strokes to filled paths.
2. Apply Pathfinder Palette>Merge. That will eliminate overlaps and union adjacent areas of same color.
3. To create a path to outline the whole thing, copy the artwork, Edit>PasteInFront, and Pathfinder Palette>Add To Shape Area. That should give you one path surrounding the whole thing. If you want that offset a bit from the arwork, try the Object>Path>OfsetPath command. Give that path a stroke color that corresponds to your cutter's recognition as a cut path and no Fill. (Depends on your cutting software and file format workflow.)