Place the egg yolk in a food processor, add the vinegar, mustard, and mashed anchovies, and process the mixture until smooth.
Alternatively, place the egg yolk in a bowl and whisk in the other ingredients with the machine running slowly add the oil through the feed tube and process the mixture until it is thick.
Or beat the oil in slowly by hand.
Beat in the water to thin the mixture slightly.
Spoon about 3 tablespoons (45 ml) of the sauce on each of 4 plates.
Sliver the endives lengthwise and spread the slices over the sauce on the plates.
Top them with the alici and sprinkle with chives.