In a stock pot, melt the butter over medium heat. Once completely melted, add the milk and salt and continue to heat until steaming. Finally, add the potatoes and simmer over very low heat until the potatoes are soft. (about 20 min)
Drain the potatoes while retaining the liquid. Clean out the stock pot and return the liquid. Using the purée blade on a food processor, purée the potato pieces and return them to the stock pot as well.
Re-combine thoroughly, adding just enough milk to make them almost pourable (like a thick pudding), and bring back to temperature.