Make the mashed potato: If you don’t have leftover mash, peel and cube 1 medium potato. Boil until fork-tender, drain well, and mash until smooth. Cool to room temperature. The mash should be thick, not watery.

Activate the yeast (if using active dry): Combine warm milk, sugar, and yeast in a bowl. Let it sit for 5–10 minutes until foamy. If using instant yeast, you can skip proofing and add it directly to the flour; still warm the milk and mix the sugar in.

Mix the dough: In a large bowl, add flour, salt, mashed potato, softened butter, one egg, and honey (if using). Pour in the milk-yeast mixture. Stir with a wooden spoon or use a stand mixer with a dough hook until a shaggy dough forms. Add 1–3 tablespoons of water if the dough seems dry; potato absorbs moisture.

Knead to smooth: Knead by hand on a lightly floured surface for 8–10 minutes or in a mixer for 5–6 minutes. The dough should be soft, slightly tacky, and elastic. It should pull away from the bowl but still feel cushy to the touch.

First rise: Place the dough in a lightly oiled bowl, cover, and let it rise in a warm spot until doubled, about 60–90 minutes. A slow, steady rise develops flavor and structure.

Portion and shape: Turn the dough out and gently deflate. Divide into 8 equal pieces for standard buns (about 95–105 g each). For sliders, make 12; for large buns, make 6. Shape each piece into a tight ball by tucking edges under and rolling against the counter for surface tension.

Flatten and proof: Place balls on a parchment-lined baking sheet, spaced a few inches apart. Gently flatten each to about 3 1/2 inches wide with your palm. Cover loosely and let rise until puffy and nearly doubled, 45–60 minutes. They should spring back slowly when poked.

Preheat and glaze: Heat the oven to 375°F (190°C). Whisk the remaining egg with a splash of water. Brush buns lightly with egg wash. Sprinkle with sesame or poppy seeds if you like.

Bake: Bake for 15–18 minutes, until deep golden on top and 200–205°F (93–96°C) inside. If tops brown too fast, tent loosely with foil near the end.

Cool and finish: Cool on a rack for at least 20 minutes. For a soft, shiny crust, rub the warm tops with a little butter. Slice only when fully warm—not hot—to avoid squashing the crumb.