The tile broke because it was not supported, that is the hard part for you to fix. Remove the broken piece, and see if there is simply a gap underneath, or if there is no pad/foundation at all to build on. If there is nothing but dirt or air underneath, then you need mortar and and enough of a base that it ties into the existing pad and is also below the frost line. This will be hard to do properly without removing additional tiles. If there is just a gap and there is something directly below to support it, and if the gap is small enough, I would use a two part epoxy to repair. I know some brands in the dual tube packages say they work for masonry on the label, and I have used them in the past for similar repairs. Epoxy is stable, resilient, and thin enough that you wont hardly notice the crack. If you get freezing winters in your area make sure there is no place for water to get in and freeze or it will only break again.
I've never done or heard of this, but most of my small tile work has been with presheeted tiles
I did read about a mosaic system where the layout was done face up and a mesh was attached, and the tile was thin-setted and afterwards grouted.
I've also seen a TOH segment where the mosaic was attached to a thin cement board (1/4"), grouted and then the whole assembly thinsetted in place.
For me, grouting from the back would mean that you couldn't know or change the amount of grout that was pushing around the mosaic. Normally, cleaning the tile of extra grout would be done in 20 to 30 minutes, before it is set solidly.
This method would let the front face be flat and the variations hidden in the back.
What would you glue the tiles with? I assume some type of water soluble glue...
Found a tutorial here this is similar to your method, called the indirect method. Do the reverse attachment to paper, install the tile on the wall (floor?), let dry and THEN grout. This makes more sense to me and overcomes what I see as a potential problem area, not being able to control the grout removal.
Best Answer
There's nothing you can do to fix it short of tearing it out and starting over. Any sanding or grinding you might try will be even more evident in the difference in finish and texture.
I doubt there is a significant difference in the thickness of the ledger panels. But even if their is, that's part of the job of a tile setter - compensating for materials and substrate to get an even, clean, finished look.
I wouldn't be happy with it either. But it will mean tearing out much of the job to fix it. Good luck.