The boxes were mounted to the plaster wall, which is behind the finished surface of the wall.
The boxes were never mounted to the studs. They were basically thrown into holes cut into the plaster wall and "secured" in whatever way was convenient.
What I found was one screw through one of the plaster ears, and a piece of bare copper wire looped through two of the other holes, and through something in the wall. There is some kind of steel mesh in the wall. Probably the ancient plaster wall had this steel mesh to strengthen it.
When the counter was installed, the finish was installed over the old plaster somehow. The distance between the old plaster wall and the finished surface is significant; Maybe not 3" as I claim, but enough to cause a problem with the box.
I was planning to mount the new boxes directly to the finished surface, which is about 3/8" thick.
If the stud was available, I would secure the boxes to them, but I have yet to see a stud.
On at least the cutout pictured, I believe the box will be smaller, and the mounting holes in the plaster ears will fall below the finished surface, causing the screws to hang freely rather than screw into the finished wall. So I need something to more or less bring the finished wall surface back to the edge of the box so that the screws can secure the box.
Perhaps some sort of flange plate that would go around the box and have holes for securing the box through the holes in the plaster ears.
Maybe what I can do is open up the cutout (to the correct height on each side of the current cutout) to fit a two-gang box and install two duplex in each.
I would end up with a cutout that is a bit high in the center, but at least I would have two of the four screws on each side of the double box into the finished wall.
Having double duplex in each location would be nice.
Each of these will have it's own 20A breaker and a GFCI outlet.
The alternate plan - double gang boxes: