In a large mixing bowl, combine the all-purpose flour, instant yeast, granulated sugar, and salt. Whisk together to ensure the dry ingredients are well mixed.
Make a well in the center of the dry mixture and add the lukewarm water, milk, and 2 large eggs. Start mixing with a wooden spoon or your hands until the dough begins to come together.
Melt the unsalted butter and let it cool slightly before adding it to the dough. Knead the dough in the bowl or on a lightly floured surface for about 10 minutes, until it’s smooth and elastic.
Place the dough in a lightly greased bowl, cover it with a damp kitchen towel, and let it rise in a warm area for about 1 to 2 hours, or until it has doubled in size.
Once the dough has risen, punch it down to release the air. Divide it into 8 equal pieces and shape each piece into a ball. Place them on a baking sheet lined with parchment paper, leaving space between each bun.
Cover the buns with a kitchen towel and let them rise again for about 30 to 45 minutes, until they puff up.
Preheat your oven to 375°F (190°C).
In a small bowl, whisk together the remaining 1 large egg and 1 tablespoon of water. Brush this mixture over the tops of the buns to achieve a beautiful golden color once baked.
Bake the buns in the preheated oven for about 15 to 20 minutes, or until they are golden brown and sound hollow when tapped on the bottom.
Remove the buns from the oven and let them cool on a wire rack before serving.
