Is there something such as an infinite potentiometer?
The device Spehro Pefhany's answer builds out of two pots is actually available as a single unit, for example the ALPS RDC803101A.

If you turn it, you get two sawtooth signals that are 180° out of phase, so when one output is in its “dead zone”, you can use the other one instead to determine the position.

This model also has no detents, so no clickyness at all.