In a large saucepan, combine the potatoes with salted water to cover generously. Bring to a boil over medium heat, cover, and simmer, stirring once or twice, until the potatoes are tender when pierced with a fork, about 25–30 minutes.
Just before the potatoes are ready, in a small frying pan over low heat, melt 1 tablespoon of the butter. Add the green onions and cook just until they turn opaque, 1–2 minutes. Remove from the heat.
Drain the potatoes, reserving about 1 cup of the cooking liquid. Return the potatoes to the pan over low heat. Using a potato masher, mash the potatoes. Using a wooden spoon, slowly stir in the hot milk, the remaining 2 tablespoons butter, the sour cream, and the green onions, and continue to stir until light and fluffy. If the potatoes seem a little dry, stir in some of the reserved cooking liquid. Season to taste with salt and pepper.
Transfer the potatoes to a warmed serving dish and serve right away.