In a small bowl, combine warm water (about 105–115°F / 40–46°C) and 1 tablespoon of the granulated sugar. Sprinkle the active dry yeast over the surface and let sit 5–8 minutes until foamy. If it does not foam, discard and start with fresh yeast.

Warm the milk and melt butter

Gently warm the whole milk until lukewarm (about 100°F / 38°C). Stir in the melted butter and let cool slightly if needed so it is not hot when combined with the yeast.

Mix dough

In the bowl of a stand mixer fitted with the dough hook (or a large mixing bowl), whisk together the all-purpose flour, remaining 1 tablespoon sugar, and salt. Add the foamy yeast mixture, warm milk-butter mixture, and the large egg. Mix on low speed until the dough comes together, then increase to medium and knead for 6–8 minutes until the dough is smooth, elastic, and slightly tacky. If kneading by hand, turn dough onto a lightly floured surface and knead 8–10 minutes.

First rise

Lightly oil a large bowl with the vegetable oil. Place the dough in the bowl, turning once to coat. Cover with plastic wrap or a clean kitchen towel and let rise in a warm draft-free place until doubled in size, about 60–90 minutes depending on room temperature.

Shape the rolls

Punch down the risen dough gently to release air. Turn it onto a lightly floured surface and divide into 12 equal pieces (about 2.5–3 oz / 70–85 g each). Shape each piece into a smooth ball by stretching the surface and pinching the seam on the bottom. Place rolls seam-side down in a 9x13-inch baking pan or two 9-inch round pans, arranging them so they touch slightly for soft sides.

Second rise

Cover the shaped rolls loosely with lightly greased plastic wrap or a towel and let rise until puffy and nearly doubled, about 30–45 minutes.

Bake

Preheat the oven to 375°F (190°C). If using egg wash, brush the tops of the rolls gently with the beaten egg and water mixture for a glossy finish. Bake in the preheated oven for 15–18 minutes, rotating the pan once halfway through, until the tops are golden brown and an instant-read thermometer inserted into a center roll reads about 190°F (88°C).

Finish

Remove the pan from the oven and immediately brush the tops with 1 tablespoon melted butter for a soft, shiny crust. Let the rolls cool in the pan for 5 minutes, then transfer to a wire rack or serve warm from the pan.

Compound Butter (Herb & Sea Salt)

While rolls are rising or baking, make the compound butter. In a small bowl, combine the softened unsalted butter, finely chopped parsley, snipped chives, lemon zest, fine sea salt, and freshly ground black pepper. Use a fork to mash and blend until evenly combined and smooth.

Taste and adjust: sample a small amount and adjust seasoning if needed (add up to another 1/4 tsp salt or a pinch more lemon zest if desired).

Shape and chill (optional): for a neat presentation, scoop the butter onto a sheet of plastic wrap, roll into a log, twist the ends and chill until firm, about 30 minutes. Alternatively, keep in a bowl at room temperature for easy spreading if serving immediately.

Serve: place the compound butter in a small dish or slice rounds from the chilled log and serve alongside the warm rolls so guests can spread while the rolls are warm.