Position a rack in the upper third of the oven and heat the oven to 350°F.
Put the beets in an 8-inch-wide, 3- to 4-quart saucepan and arrange snugly. Add the butter, sugar, 1/2 tsp. salt, and enough water to just cover the beets (about 2 cups). Bring to a boil over high heat.
Cook over high heat, shaking the pan occasionally, until the liquid has reduced to a syrupy glaze and the beets are tender, about 10 minutes. (If the glaze is done before the beets, add about 1/2 cup water and continue to cook. If the beets are done first, remove them and continue to boil the liquid until syrupy.)
Meanwhile, spread the walnuts on a small rimmed baking sheet and toast in the oven until fragrant and a few shades darker, about 8 minutes.
Lower the heat under the beets to medium low, add the warm walnuts, and toss to combine. Season to taste with salt and serve sprinkled with the chives. (The glazed beets and walnuts, without the chives, can be kept warm, covered, for about 20 minutes.)