In a large saucepan, combine the potatoes, 1 teaspoon salt and cold water to cover. Bring to a boil over high heat. Lower the heat to maintain a simmer and cook until fork-tender, about 10 minutes. Drain the potatoes, reserving 1/2 cup of the cooking liquid. Return the potatoes to the pan. Shake the pan over medium heat for about a minute to dry the potatoes. Transfer the potatoes to a food mill, ricer or bowl.
Add the oil and reserved cooking liquid to the saucepan and warm over medium heat. Remove the pan from the heat. Mash the potatoes through the food mill or ricer into the pan (or mash by hand in the bowl and transfer to the pan). Add the basil, parsley and tarragon and stir to combine. Season with salt and pepper. Serve immediately.
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