The method I use for fragging the majority of hard corals is primarily the same with the cutting tooling being an Inland band saw.
Cooling liquid used is fresh mixed saltwater with enough iodine to color it a light amber. This helps to disinfect cuts as they're made which has shown to greatly increase frag survival.
All corals are stored during cutting in a small bucket holding water taken directly from their home aquarium. This water is used both to keep them wet as well as for rinsing any flesh away from cuts while I'm working on them.
All finished, and rinsed, frags or trimmed colonies are soaked in Brightwell Aquatics Restor dip to insure that minimal flesh is lost.
Both soak buckets are rinsed and replenished between colonies to reduce the risk of interactions between loose flesh of different coral species.
Notes about Chalice (Echinophyllia):
Chalice corals have some of the most delicate of skeletal structures which means that extreme care must be taken throughout the process from the time of cutting to when frags are secured to plugs or other mounts. Many of the structures within the flesh are free floating and must be maintained in their original position to guarantee the survival of the coral.
Always enter the edge of colonies at a perpendicular angle to insure that full skeletal blades are kept intact within the flesh.
Value of chalice frags is often set by the quantity of eyes included, so don't devalue them by cutting them in half.
As with all LPS corals, survival and growth rate after fragging can be greatly increased by avoiding as many polyps as possible while cutting.