I was taught by my dad to cut the fish and remove the head (behind the gills 1/2 inch or so).
That is not what I am talking about with bream throats.
What I am describing is cutting the fish at the bottom of the gill, twisting the head up, and cutting the top of the gill and backbone. The fish will be left with the rear gill opening still on it. Not a shred of meat is wasted.
I was halfway kidding about the bream throats, but that is what they are, actually. Just not cut out of the fish like snapper/grouper.