Find the receptacle or GFCI you want to use first. 314.24(B)(4) requires boxes to be 1/4" deeper than the depth of the mounted device if the device is 1" or deeper from its mounting yoke. It also specifies a minimum box depth of 1 3/16 ". This applies only to #10 and #12 circuits. But if this is a kitchen counter/island, it will most likely be #12. I think the idea of shortening the top drawer is a good one if that works. Many times, the drawer backs are a few inches from the back of the cabinet anyway. You may be able to squeeze a deep single gang box in that cavity with the factory drawer depth.
Another solution - an outlet on the counter fed from below. Like a tombstone box near the back of the cabinet. Now someone can sit at that overhang, eat, and charge their phone...