Steam the potatoes in ½ inch of water, covered, until tender, 18 to 20 minutes (adding more water to the pot if necessary); drain.
Meanwhile, heat ½ cup of the butter in a medium skillet over medium heat. Cook, stirring occasionally, until the butter foams and begins to brown, 6 to 8 minutes.
Mash the potatoes with the milk, 1½ teaspoons salt, and ¾ teaspoon pepper. Fold in the remaining ¼ cup of butter. Top the potatoes with the brown butter and chives. Serve warm.