Scrub and peel the potatoes, then cut them into ~1" chunks.
Add the potatoes to a pot large enough to hold them with at least 4" of clearance from the top. Add water to cover the potatoes by at least 2" and add a palmful of salt.
Bring the water and potatoes to a boil and then reduce to a light boil. Continue boiling until a cake tester (or paring knife) meets no resistance when inserted into a potato chunk (~15 to 20 minutes).
Reserve 1/2 cup of the cooking water, then drain the potatoes. Melt the butter in the pot; add the potatoes followed by a pinch of salt. Use a fork to mash and adding the reserved water as need to the reach the desired smoothness.
Season to taste with salt.