Bring potatoes, salt and 6 cups water to a boil in a 4-quart saucepan over medium-high heat, stirring occasionally to ensure the salt has dissolved. Lower to a simmer, then cook until potatoes are tender, 18 to 20 minutes.
Pour the water and potatoes into a colander in the sink and leave to air-dry for 5 minutes; you will see the salt start to form on the outside of the potatoes.
Transfer the potatoes to a serving dish, then pour the melted butter and sprinkle the chives over the top. Toss together, then serve immediately.