Heat the oil in a small saucepan over medium-low heat. Add the potato and onion. Cook, stirring, without browning, for 10 to 15 minutes, until very tender; set aside. Bring a large pot of salted water to a boil and place a bowl of ice water on the side. Separate 8 of the largest leaves from the lettuce heads. Boil for 5 seconds, strain, and chill in the ice water. Being careful not to tear, remove the leaves from the ice water, pat dry, and reserve for the tartar. Separate the remaining leaves from the stems and boil them for 20 seconds, then strain and chill in the ice water. Squeeze dry and transfer to a blender with the cooked potato and onion. Puree until smooth, pass through a fine-meshed sieve into a bowl, and chill over ice. If needed, adjust the seasoning with salt and reserve, chilled.