Place the peeled and cut potatoes in a large pot. Cover with cold water by about 1 inch and add 1 tablespoon kosher salt.
Bring to a boil, then reduce the heat to maintain a gentle boil. Cook for 15–20 minutes, or until the potatoes are fork-tender.
Drain the potatoes well and return them to the hot pot. Place over low heat for 1–2 minutes, gently tossing occasionally, to allow excess moisture to evaporate. Remove from the heat.
Add the butter to the hot potatoes and mash until the butter is melted and incorporated.
Add the sour cream, 1 teaspoon kosher salt, and black pepper. Mash gently until combined.
Gradually add the warm milk, mashing just until the potatoes are creamy. Avoid overmixing.
Taste and adjust the salt and pepper if needed. Transfer to a serving bowl and top with additional butter, if desired. Serve warm.