Another of the classic "mother sauces", (Sauce) Mayonnaise, the old-school Sauce Allemande, the popular Hollandaise and Bearnaise sauces, and Suace Colbert are all examples of egg-thickened sauces.
These are popular with the low-carb crowd, but they are excellent sauces in their own right.
Unlike the White Sauce and Veloute that rely on flour as a thickener, the egg-thickened sauces use (wait, wait! Don't tell me...) egg yolks for thickening. Egg yolks can absorb a quantity of fat into an emulsion similar to how flour and fat absorb liquid. But the egg yolk-fat emulsion has no stock or other liquid, so it is very rich and is often seasoned very strongly.