Place the potatoes in a large pot or Dutch oven. Add the chicken broth and enough water to cover them by about 1- to 2-inches. Place over HIGH heat and bring to a boil. Reduce heat, cover, and simmer for 20 to 25 minutes, or until very tender all the way through when pierced with a fork or the tip of a sharp knife.
Meanwhile, allow the sour cream to rest at room temperature while the potatoes simmer. Warm the milk in a saucepan over LOW heat or in the microwave, just until warm. Set it aside.
Drain the potatoes well, return them to the warm, empty pot and set it over LOW heat. Heat, tossing the potatoes, for a minute or two to evaporate any remaining moisture and let the potatoes dry a bit. Remove the pot from the heat.
Use an electric hand mixer on LOW speed to break up the potatoes a little then add the butter, sour cream, and the salt. Beat again on MEDIUM speed, adding the warmed milk as you mix. Add just as much milk as needed to reach the desired consistency and mix just until creamy and smooth. Avoid overworking the potatoes as it can result in a gummy, gluey texture. Taste and add additional salt, if needed.
Serve as is or garnish with chives or parsley and freshly ground black pepper, if desired.