Tile – cut tile to follow the profile of a closet flange

You just want a stable platform for your tile. It depends on the size of the tile.. If you are cantilevering 10-15% of a larger tile, just fill the thinner areas with more mud and put backer where it will fit.

It looks like you needed some blocking behind the green board (where the rear curve comes in) to be able to fill in. Perhaps some fiberglass mesh tape would let you land some thinset to bridge the gap.

A diamond hole saw will do the job, use plenty of water and light pressure.

You will do better with a single hole, AFA breakage goes, compared to making 2 edge semi circles.

Update

If we are talking about a toilet flange, there are 2 methods, each with variations, depending on your equipment.

1. With a tile wet saw, raise and lower the blade such that you cut diameter lines edge to edge. Rotate the tile 10 degrees and repeat. Flip the tile over and repeat from below. Remove teeth with an angle grinder fitted with a diamond blade. If you are connected to a GFCI outlet, a helper can slowly spritz the cutting with a water bottle. It will keep the dust to a minimum. I would recommend a continuous edge (not segmented or stepped) diamond blade for this. The cheap table type tile wet saws actually excel for this cut, allowing fast and accurate cuts, all from below.
2. The whole cut can be made with an angle grinder setup described above, you just have to tip the blade at 45 to the surface of the tile and sneak up on the line from the inside of the desired cut out. With marble you have to go slow and minimize pressure. This method also works nicely for electrical boxes.