Pull-Apart Potato Rolls

Boil potato in a small saucepan of boiling water (no need to add salt) until a paring knife passes through flesh with no resistance, 30–40 minutes; drain. When cool enough to handle, pass through ricer into a small bowl (peel won't go through; discard).
Mix milk and 3/4 cup riced potato in the bowl of a stand mixer with whisk attachment until no lumps remain. Add 1/2 cup butter and mix until incorporated. Switch to dough hook. Add all-purpose flour, yeast, and sugar and mix on medium speed, scraping bottom and sides of bowl as needed, until a very wet, sticky dough forms, about 2 minutes.
Let dough rise, uncovered, in a warm spot, 30 minutes (it will have puffed slightly).
Add eggs, egg yolk, 2 2/3 cups bread flour, and 1 Tbsp. salt and mix on medium-high, adding more bread flour if needed, until dough is smooth and elastic, about 5 minutes. Brush surface of dough with butter, cover, and let rise in a warm spot 30 minutes (dough should rise 1 1/2 times its initial size).
Turn out dough onto a lightly oiled surface and divide into 18 pieces; roll each into a ball using your palm. Brush a 13x9" baking dish with butter and place balls side by side in dish (rolls will be touching). Brush tops with more butter. Let sit, uncovered, in a warm spot 1 hour.
Preheat oven to 400°F. Brush dough again with butter and sprinkle with salt. Bake rolls until deep golden brown, 15–20 minutes. Transfer dish to a wire rack and let rolls cool in dish 10 minutes. Turn out rolls onto rack and let cool 30 minutes before serving.