This recipe calls for using a potato ricer. When using a potato ricer, scrubbed potato may be boiled with skin on and the ricer will remove the skin. If you do not have a potato ricer, peel the potato, boil and mash well until no lumps remain. Boil the potato until a paring knife passes through flesh with no resistance, about 30-40 minutes; drain. If mashing, do so immediately; if using a ricer, wait until the potato cools enough to handle and pass through the ricer into a small bowl and discard peel.
Using whisk attachment on a stand mixer, mix milk and 3/4 cup mashed/riced potato until no lumps remain. Add 1/2 cup butter and mix until incorporated. Switch to dough hook and add all purpose flour, yeast, and sugar; mix on medium speed, scraping bottom and sides of bowl as needed. A wet sticky dough will form in about 2 minutes. Let the dough rise uncovered in a warm draft free place for 30 minutes, it will have puffed slightly. Add eggs, egg yolk, 1 Tbsp. salt and 3 cups bread flour; mix on medium high, adding more bread flour as needed, until dough is smooth and elastic, about 5 minutes. Brush dough surface with butter, cover and let rise another 30 minutes in a warm spot; dough should rise 1-1/2 times its initial size. Butter a 13x9-inch baking pan and lightly oil countertop.
Turn dough out; divide into 18 equal pieces and roll each into a ball. Place balls in baking pan with sides touching and brush with more butter. Let sit, uncovered, in a warm spot for 1 hour.
Preheat oven to 400°F. Brush rolls with butter again. Bake rolls until deep golden brown, 15-20 minutes. Cool rolls in pan on wire rack 10 minutes then turn rolls out onto the rack and cool another 30 minutes before serving.