Warm the milk to about 110°F. Stir in the sugar and yeast and let it get foamy, 5–10 minutes.
Mix in the mashed potatoes, melted butter, and salt until mostly smooth.
Add 3 cups of flour and stir until a shaggy dough forms. Rest 10 minutes to hydrate.
Knead on a lightly floured surface, adding just enough extra flour for a soft, smooth dough, 8–10 minutes.
Grease a bowl, place the dough inside, and turn to coat. Cover and let rise until doubled, 60–75 minutes.
Grease a 9×5-inch loaf pan. Gently deflate dough, shape into a tight loaf, and set it into the pan seam-side down.
Cover and let rise until the dome is about 1 inch above the pan, 30–45 minutes. Preheat oven to 375°F.
Bake until deep golden and 195–200°F inside, 30–35 minutes. Tent with foil if browning too fast.
Brush the hot loaf with melted butter. Cool in the pan 10 minutes, then transfer to a rack to cool before slicing.