Peel and cut the potatoes in 1/2-inch slices. Rinse under cold water and transfer to a large saucepan. Add cold water to cover the potatoes by an inch and add the salt. Set over medium-high heat, cover, and bring to a simmer. Remove the lid, lower the heat to medium, and cook for 15 to 20 minutes, until tender.
While the potatoes cook, mash the anchovies, garlic, pepper, and 1 tablespoon of the olive oil with a mortar and pestle, or with a bowl and fork. Combine the anchovy mixture with the cream and warm up for a few seconds in the microwave (or in a small saucepan over medium heat) until hot but not boiling.
Drain the potatoes and return to the pan. Add the anchovy mixture and the remaining 3 tablespoons olive oil. Set the pan over medium-low heat, and mash the potatoes with a fork or potato ricer, incorporating the other ingredients. Taste, adjust the seasoning, and keep warm until ready to serve.