Step 1
Make the mayonnaise: In a large bowl, whisk together the mustard, vinegar, salt, white pepper, and egg yolks. Whisking continuously, very gradually add the oil, starting with a few drops at a time, until all the oil is incorporated and a thick sauce forms. (Alternatively, in a food processor or blender, pulse together the mustard, vinegar, salt, white pepper, egg yolks, and 1 tablespoon of cold water. With the motor running, in a slow, steady stream, add the oil and blend until thick.) Scrape into a bowl, cover, and refrigerate until ready to use.
Step 2
Make the eggs: Into a medium pot of boiling water, gently lower the eggs. Bring the water back to a boil, then turn the heat to medium-high and simmer for exactly 9 minutes. Pour out the hot water and shake the pan to crack the eggshells. Fill the pot with cold water and ice and set aside to cool.
Step 3
In a small bowl, stir together the ketchup, a few dashes of hot sauce (or to taste), 3 tablespoons of the reserved mayonnaise, and cold water as needed to make a thick, pourable sauce.
Step 4
Shell the eggs and halve them lengthwise. On four plates, make beds of lettuce if desired. Divide the eggs evenly cut-side down among the plates, then spoon evenly with the ketchup-mayo sauce and top with the anchovies, capers, and onion.