Attached is a picture of two countersinks. They are unusual in that they
have a pilot shaft, but it is not a drill. To use them, one has to drill a
pilot hole and then use the countersink. Because they have a shaft they
stay nicely centered and because they lack a pilot drill they are not
pulled down into the wood giving very nice control of depth.

They do get quite warm though because of the friction of the shaft against
the walls of the pilot hole.

These belong to a friend. He does not remember where he got them but both he
and I would like to know where to buy countersinks like this (and what they
are called).