In a medium bowl, mash up the hard-boiled and raw egg yolks with the mustard and anchovy fillet (if using). Slowly begin drizzling in the oil a tablespoon or so at a time while whisking continuously, being sure that all the oil has been emulsified into the yolks before adding more. Once all the oil has been added and you have a very thick and creamy emulsion, whisk in the vinegar. Fold in the cornichons, capers, and parsley. Finely chop the egg white and fold that in too. Season to taste with salt, black pepper, and cayenne, then use immediately or transfer to an airtight container and refrigerate for up to 3 days.