For the mayonnaise, blend the egg yolks, Dijon mustard, salt, pepper and lemon juice in a food processor until well combined. Season, to taste with salt and freshly ground black pepper.
With the motor still running, gradually add the olive oil in a thin stream until all of the oil is well combined into the mixture. Repeat the process with the sunflower oil. If the mixture becomes too thick, loosen it with a tablespoon or two of warm water and blend again until smooth and well combined.
Transfer the mayonnaise to a bowl and add the chopped anchovy fillets, if using, and the baby capers. Stir well to combine. Set aside.
For the potato salad, bring the potatoes to the boil in a large pan of cold, salted water. Reduce the heat to a simmer, then continue to simmer for 15-20 minutes, or until tender. Drain well and refresh under cold running water until cold.
Half-fill a large separate pan with cold water, add the quails' eggs and bring to the boil. Continue to boil for two minutes, then drain well and refresh under cold running water until cold.
Slice the cooked potatoes and transfer them to a large serving bowl. Peel the quails' eggs, cut them in half and add to the bowl of potatoes. Sprinkle over the onion, tarragon, chives and parsley, then mix well to combine.
Spoon the mayonnaise on top of the salad and mix using two large serving spoons until all of the salad ingredients are evenly coated in the mayonnaise. Season, to taste, with salt and freshly ground black pepper. Serve immediately.