Put the potatoes in a large saucepan, cover with cold water by about 1 inch and add 2 tablespoons salt. Bring to a boil, uncovered, over high heat, and then reduce the heat to medium-high and simmer briskly until fork tender, about 45 minutes. When the potatoes are almost done, heat the milk in a small saucepan over medium-high heat to just under a simmer. Set aside.
Drain, and then halve the potatoes. Pass them through a ricer or food mill (remove the skins as you go) back into the pan. Sprinkle the vinegar over the potatoes. Return the pan to low heat and fold in the warm milk. Fold in the butter a few bits at a time until fully incorporated. Season with salt. Fold in the dill. Serve hot.