For the remoulade: Chop all the ingredients together and add them to the mayonnaise in the food processor; pulse a few times for them to blend. Refrigerate for 30 minutes before serving.;
First make the mayonnaise: In a non-reactive bowl or in a food processor, combine the egg yolks, mustard, lemon juice, and water. Whisk or pulse with the machine to break up the yolks.
If you're making the mayonnaise by hand, put the bowl on a damp towel to keep it from moving around while you work. Then drizzle in the oils, whisking constantly, to form an emulsion. If the mayonnaise breaks, stop drizzling and whisk until it comes together again. If you're using the food processor, pour in the oils in a thin stream, with the machine running. Taste and adjust seasoning with salt and pepper and more lemon juice.