Boil the peeled potato in water until completely soft, about 20 to 40 minutes depending on size, then mash it through a fine mesh strainer until silky smooth.
In a small bowl, warm the milk and water to 90–95°F, whisk in the yeast and sugar, and let it sit for about 2 minutes until foamy.
In a stand mixer bowl, whisk together the bread flour, salt, turmeric, and optional amylase, then add in the yeast mixture, oil, egg, egg yolk, and mashed potato and mix on low for 5 minutes until smooth and elastic.
Add the softened butter and continue mixing for another 2 to 3 minutes until the dough becomes shiny, sticky, and fully pulled together.
Grease your hands and a bowl, form the dough into a rough ball, place it in the bowl, cover with plastic wrap or a damp towel, and let it rise for 1½ to 2 hours until doubled in size.
Punch down the dough, turn it out onto a work surface, divide into 10 to 12 pieces (about 105g each), shape into balls, and gently flatten each one.
Place the dough balls seam-side down on greased parchment-lined baking sheets, flatten slightly with the bottom of a glass, cover loosely, and let them rise again for 30 to 60 minutes until puffed.
Brush the tops lightly with milk and bake at 375°F (191°C) for 15 to 17 minutes, until golden brown.
Immediately brush the tops with melted butter and cool the buns completely on a wire rack before slicing or storing.