Peel the eggs, halve, and separate the whites from the yolks.
Place the egg yolks in a medium bowl and mash until they resemble cornmeal (some pea-size pieces will remain).
Add the mustard and continue mashing until it is well incorporated.
Add the oil to the yolks, pouring in a thin, steady stream while constantly whisking to form a paste.
Stir in the vinegar, salt, and pepper.
Finely chop the egg whites and add them to the yolk paste.
Add the remaining ingredients and fold everything together until evenly combined.
Taste and season with additional salt and pepper as needed.