Bring a large saucepan of water to a boil over high heat and fill 2 medium bowls with ice water. Carefully lower eggs into saucepan using a slotted spoon or spider skimmer, and boil for 30 seconds. Cover, reduce heat to low, and simmer, undisturbed, for 7 minutes. Transfer 3 eggs to 1 bowl of ice water. Continue simmering remaining 3 eggs, covered, for 3 minutes more; transfer to second bowl of ice water. Let eggs cool completely, about 15 minutes.

Peel jammy 7-minute eggs under cool running water, and halve lengthwise. Spoon yolks into a large bowl; chop egg whites, and set aside. Add Dijon, hot sauce, Worcestershire, salt, pepper, and paprika to bowl with egg yolks, and whisk until completely combined with no yolk chunks remaining, about 1 minute. Slowly stream in olive oil and canola oil, whisking constantly, until emulsified. Set aside.

Peel remaining 3 hard-cooked eggs, and grate on large holes of a box grater. Add to large bowl with yolks and oil mixture along with chopped cornichon, chives, and reserved egg whites; fold until combined. Transfer to a serving bowl and garnish with additional black pepper, paprika, and chives. Serve with crudités, potato chips, or crackers.