If using the nuts, preheat the oven to 350°F Spread the nuts on a baking sheet and place in the oven until toasted and fragrant (8-10 minutes). Remove from the oven, let cool and chop. Set aside.
Meanwhile, bring a saucepan three-fourths full of water to a boil over high heat. Add the potato pieces and boil until tender when pierced with a fork, about 15 minutes. Drain well, return the potatoes to the pan and place over high heat for 1-2 minutes to evaporate the moisture, turning them to prevent scorching. Remove from the heat and, using a potato masher, mash the potatoes until smooth. Set aside.
In a mortar, combine the garlic with a little sea salt and mash with a pestle until pureed. You should have about 2 tablespoons of pureed garlic.
Stir the garlic into the potatoes and, using a fork, beat in 1 tablespoon of the vinegar, half of the lemon juice and half of the olive oil. Transfer to a food processor fitted with a the metal blade. With the motor running, gradually add the remaining 2 tablespoons vinegar, lemon juice and olive oil (and nuts if using.) Season to taste with more vinegar if needed, and salt and pepper.
Transfer to a bowl and serve. If refrigerating, bring to room temperature before serving.