Cook peeled and diced potatoes in boiling salted water until very tender, about 12–15 minutes. Drain thoroughly and return to the pot over low heat for 1–2 minutes to dry. Mash until smooth and measure 1 1/2 cups. Cool to warm (not hot).

Activate yeast

Warm the milk to about 105–115°F (40–46°C). Stir in 1 tablespoon of the sugar from the total and sprinkle the 2 tsp instant yeast over the surface. Wait 3–5 minutes until foamy. (If using active dry yeast, proof 5–10 minutes.)

Mix dough

In the bowl of a stand mixer fitted with the dough hook (or in a large bowl), combine 3 1/2 cups of the flour, remaining sugar (2 tbsp), and 1 1/2 tsp salt. Add the mashed warm potatoes, foamy yeast-milk mixture, 1 large egg, 1 tbsp vegetable oil and 2 tbsp softened butter. Mix on low until ingredients come together, scraping bowl as needed.

Knead and incorporate butter and flour

Gradually add remaining 1/2 cup flour as needed while kneading on medium for 6–8 minutes (by mixer) or 10–12 minutes by hand, until dough is smooth, slightly tacky but not sticky. Add remaining 2 tbsp softened butter about halfway through kneading so it fully incorporates. Dough should be elastic and pass the windowpane test.

First rise

Lightly oil a large bowl and place dough inside, turning to coat. Cover with plastic wrap or a damp towel. Let rise in a warm draft-free place until doubled in size, about 60–90 minutes depending on room temperature.

Shape & proof

Divide dough

Gently deflate the risen dough and turn it onto a lightly floured surface. Divide into 12 equal portions (about 75–80 g / 2.6–2.8 oz each) for medium buns. Shape each portion into a tight ball by stretching the surface and tucking seams underneath.

Arrange on baking sheet

Place shaped buns on a parchment-lined baking sheet or in two 9-inch round cake pans (for pull-apart buns), spacing about 1–1.5 inches apart. Cover loosely with lightly greased plastic or a kitchen towel for the second rise.

Second rise (proof)

Let the buns rise until puffy and nearly doubled, about 30–45 minutes. They should spring back slowly when pressed lightly.

Bake & finish

Preheat oven

About 15 minutes before baking, preheat the oven to 375°F (190°C) with a rack in the center.

Egg wash

Whisk together 1 large egg and 1 tsp water. Brush the tops of the risen buns lightly with the egg wash, taking care not to deflate them. Sprinkle optional coarse sea salt or sesame seeds if desired.

Bake

Bake for 15–18 minutes (for individual buns) or 20–25 minutes (if using a pan), until tops are deep golden brown and internal temperature reaches about 190–195°F (88–90°C). Rotate the pan halfway through baking if your oven bakes unevenly.

Cool

Remove buns from oven and brush lightly with 1 tbsp melted butter if you like a soft, glossy crust. Transfer to a wire rack and cool at least 15 minutes before serving. For best texture, consume within 2 days or freeze cooled buns in an airtight bag.