Put the potatoes in a medium saucepan and cover with cold water by 2 inches and season generously with salt. Bring to a boil over high heat, lower the heat to maintain a gentle simmer and cook until very tender, about 30 minutes. Drain the potatoes and let cool slightly.
Rub the skins off the potatoes and discard them. Coarsely chop the potatoes and puree them through a food mill or ricer into a medium bowl.
Meanwhile, on a cutting board, lightly sprinkle the garlic with a generous pinch of the salt and smash it into a fine paste with the side of a cook's knife.
In a food processor, combine the garlic, almonds, and oil and puree into a paste. Mix the oil mixture into the potatoes until incorporated; then mix in the 1 tablespoon and 1 teaspoon salt, water, lemon juice, and vinegar and season with pepper, to taste. Serve.
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