Melt 2 tablespoons of the butter in a large soup pot or Dutch oven over medium heat, then add the chopped onion and a big pinch of salt. Cook, stirring frequently, until the onion is translucent, about 5 minutes.
Add the peeled and chopped potatoes to the pot, then stir in the broth and 1 teaspoon of salt. Increase the heat to medium-high, bring the broth to a bubble, and cook for 10 to 15 minutes, until the potatoes are tender. (Check them with a sharp paring knife or fork: When the knife or fork can go all the way through one of the bigger potato pieces without resistance, you can move to the next step.)
Remove the pot from the heat. If you have an immersion blender, use it to puree the soup right in the pot. Otherwise, you can puree the hot soup in batches in a blender or food processor, then return it to the pot.
Cut the remaining 4 tablespoons of butter into small pieces and add them to the pureed soup. Stir until the butter is completely melted, then taste the soup and add more salt and a few grinds of black pepper as needed/desired.
Ladle the hot soup into bowls, and serve it with an assortment of toppings at the table.