The Robert Sorby Mushroom Tool (ref. 804H) has been specifically designed for under-cutting rims on turned projects. The HSS tip give sufficient clearance for a cleanly feature.

Other tips can also be fitted to the same shank -

Mini Round Nose Cutter (ref. 804C06)
Rim Cutter (ref. 804C05)
Side & Face Cutter (804C07)
Side & End Cutter (804C08)

Tips are sharpened using a diamond file or hone.