Peel the potatoes and cut them into 1/2-inch chunks.
Add the potatoes and chopped onion to a large pot. Pour in enough water to just cover the potatoes, then stir in the salt.
Bring to a boil over medium-high heat. Reduce heat and simmer uncovered for 12–15 minutes, or until the potatoes are very tender.
Drain off most of the water, leaving about 1/2 cup in the pot.
Stir in the butter until fully melted.
Pour in the milk and heat gently over low heat, lightly mashing some of the potatoes to create a creamy texture. Do not let the soup boil.
Season with black pepper and additional salt if needed. If the soup is too thick, add a splash more milk; if too thin, simmer a few extra minutes.
Ladle into bowls or mugs, top with a little butter if desired, and serve warm.
Old Fashioned Potato Soup
