Combine warm water (about 100–110°F / 38–43°C) and warm milk in a small bowl or measuring cup. Stir in granulated sugar and sprinkle the active dry yeast over the surface. Let sit 5–8 minutes until foamy. If it doesn't foam, yeast may be inactive — use fresh yeast.

Mix dry ingredients

In the bowl of a stand mixer fitted with the dough hook (or a large mixing bowl), combine the all-purpose flour and 1 tsp fine sea salt. Whisk briefly to distribute.

Combine wet and dry

Add the foamy yeast mixture, melted butter (4 tbs), and the large egg to the dry ingredients. Mix on low speed until ingredients come together (about 1–2 minutes). Increase speed to medium-low and knead for 6–8 minutes until the dough is smooth, slightly tacky but not overly sticky. If mixing by hand, knead on a floured surface 8–10 minutes.

First rise

Lightly oil a clean bowl and place the dough inside, turning once to coat. Cover with plastic wrap or a damp towel. Let rise in a warm draft-free spot until doubled in size, about 60–75 minutes.

Shaping, Filling & Baking

Prepare butter filling

While dough rises, soften 6 tbs unsalted butter to room temperature. In a small bowl, mash the butter with 1.5 tsp fine sea salt until smooth and evenly salted. Set aside at room temperature so it remains spreadable.

Roll out dough

Once risen, gently deflate the dough and turn it out onto a lightly floured surface. Roll into a rectangle about 12 x 10 inches and roughly 1/4 inch thick.

Spread butter

Use an offset spatula or the back of a spoon to evenly spread the softened salted butter over the dough, leaving a 1/2-inch clean border on one long side to help seal.

Roll and portion

Starting from the long edge opposite the clean border, tightly roll the dough into a log. Pinch the seam to seal. With a sharp knife or bench scraper, cut the log into 10 equal pieces (about 1 to 1¼ inches each). Place the cut rolls seam-side up in a greased 9x13-inch baking pan or two smaller pans, leaving small gaps for expansion. Alternatively, you can place them seam-side down for a smoother top.

Second rise

Cover the pan lightly with plastic wrap or a damp towel and let the rolls rise until puffy and nearly doubled, about 30–45 minutes. Meanwhile preheat the oven to 375°F (190°C) about 15 minutes before baking.

Egg wash and salt finish

Brush the tops gently with the beaten egg wash. Sprinkle the tops lightly with coarse flaked sea salt (about 1 tsp distributed across the rolls) to give noticeable salt crystals and texture.

Bake

Bake in the preheated 375°F (190°C) oven for 18–22 minutes, or until the rolls are deep golden brown on top and an internal temperature near the center measures about 190–200°F (88–93°C). Rotate the pan halfway through baking if your oven has hot spots.

Finish

Remove from oven and optionally brush with 1 tbs melted butter for extra sheen and richness. Let cool 10 minutes in the pan, then transfer to a wire rack or serve warm straight from the pan.