Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 15 to 20 minutes.
Drain and transfer to a bowl; mash until smooth. Set aside to cool.
Put garlic, salt, and lemon juice in a blender or food processor; pour in enough olive oil to lightly coat garlic cloves. Blend mixture on high; slowly stream remaining olive oil through the top opening of the blender or food processor until mixture is smooth.
Add mashed potatoes, about 2 to 3 tablespoons at a time, to the garlic mixture while blender is still running. Continue adding potatoes until all are incorporated.
Transfer sauce to a bowl and refrigerate until chilled, 1 hour to overnight.
