- In a medium pan, bring half a pan of water to a boil.
- Add potatoes, cover, and cook over medium heat until potatoes are just tender when pierced, 30 to 40 minutes.
- Remove all but one potato from cooking water; immerse in cold water.
- When cool, cut potatoes into 1/2-inch-wide slices.
- Continue to cook remaining potato over medium heat until very soft when pierced, and skin has begun to peel away, 5 to 10 minutes.
- Remove potato and cut in half; slice 1 half and add to the other sliced potatoes.
- Immerse remaining half in cold water until cool to touch, about 3 minutes; peel, cut in half again, and place in a blender with 1/2 cup of the cooking water.
- Whirl until smooth.
- Place in a small bowl; immerse bowl just to the rim in cold water until potato puree is cool.
- In a medium bowl, combine minced garlic, Worcestershire, Dijon mustard, lemon juice, red wine vinegar, and minced anchovies.
- Mix in 200ml of the cooled potato puree; pour over sliced potatoes. (Discard remaining puree or reserve for another use.)
- Add grated parmesan cheese; gently mix to coat potatoes.
- Garnish with minced parsley and whole anchovy fillets.
- Add salt and pepper to taste.
Serve, or if making ahead, chill up to 1 day.