Rinse and brush potatoes thoroughly, cut into 2 pieces if sizes are too varied. Leave skin on.
In a pot, bring water and tablespoon of salt to boil over high heat. Add potatoes and cook until soft, about 15 minutes (check tenderness with fork).
Drain the potatoes in a colander. While the potatoes are still hot, press them through a fine-mesh potato ricer or food mill.
If equipment is not available, then press them through a fine-mesh potato ricer or sieve into a large sauce pan.
Discard any skin that will gather in the ricer or sieve. Process the potatoes two more times through the ricer or sieve to ensure a fine texture.
Separately, bring milk to a boil in a medium saucepan over medium-high heat.
Place the saucepan with the pureed potatoes over low heat.
Once it has heated up until some of the moisture has evaporated, slowly stir in the butter until completely incorporated. Whisk in the milk next, using a large whisk.
Pass potato puree through ricer or sieve one final time before serving for an even lighter texture. Serve in mini cocotte.