Bring 3 quarts (2.8L) water to a boil in a large pot and prepare an ice bath. Carefully lower eggs into pot and continue to boil for 30 seconds. Cover tightly, reduce heat to low (water should maintain a bare simmer), and continue cooking for 9 minutes for jammy yolks or 11 minutes for hard-cooked yolks (see headnote above for guidance on how the degree of yolk doneness can affect the sauce). Immediately transfer eggs to ice water and allow to cool for at least 15 minutes, then peel under cool running water. Halve eggs lengthwise and scoop yolks into a medium mixing bowl, blender, or food processor. Finely dice egg whites.

Add vinegar, mustard, and a pinch of salt and pepper to the egg yolks. Whisk, blend, or process until yolks are almost entirely combined with the other ingredients and very few if any yolky lumps remain. While continuously whisking or blending, slowly drizzle in oil in a thin, steady stream. The sauce should emulsify, but it may break more easily with hard-cooked yolks (see headnote above on why you may or may not want this); you can also whisk less vigorously and/or add the oil much faster to deliberately create a broken sauce.

Stir in cornichons, capers, herbs, and egg whites until well combined. Season with additional salt and pepper, if desired, then serve or refrigerate in an airtight container until ready to use.