Designing Kitchens and Bathrooms - What is the procedure for reattaching broken ceramic countertop tile?

The curved corner tile (similiar to an "AN-1662") of my '50's sink counter has broken off. :( It came off cleanly in one piece along the groutline with the exception of a small fraction of it which remains adhered to the counter top. All the grout is intact on the broken piece . Since this was a 50's installation finding a replacement tile is out of the question. I just want to reattach the broken tile. Also it would be impossible to remove the "mud" or whatever it's called that the tile was set in.

Can the piece - including the grout - just be glued back?

If yes, what kind of glue and how do I go about it?

Do I need to apply pressure as it dries - and if so, how would I do that (as it is the corner piece)?

If the grout needs to be removed from the broken piece, how would I do this without further damaging the piece?

(I'd prefer to keep the grout on as trying to match it would also be impossible - not to mention expensive considering it's only one piece).

Any help in this would be appreciated.

A corner piece away from the usual wet and traffic flow could be reset with some silicone caulk. If the grout is white, caulk could also be used for that.

Just confirm if the materials are OK to be used in kitchens / wet areas