Place the potatoes and 1 teaspoon of the salt in a medium saucepan, add enough water to cover, and bring to a boil over high heat. Reduce the heat to moderate and simmer until the potatoes are tender when pierced with a fork, about 45 minutes. Drain, reserving 1/4 cup of the cooking water.
Return the potatoes to the pan and set over low heat, uncovered, for about 5 minutes, stirring occasionally, to let the potatoes dry out a little (too much moisture will dilute their flavor). For the smoothest potatoes, pass them through a food mill. For a slightly coarser puree, mash with a potato masher or fork or use a hand mixer.
Beat the buttermilk into the potatoes with a wooden spoon until thoroughly incorporated. If you prefer even creamier potatoes, add a little of the reserved cooking liquid. Beat in the butter, the remaining 1/2 teaspoon salt, and plenty of pepper.
Serve at once, or keep the potatoes warm, covered, in a double boiler over hot water for up to 1 hour.