In a large bowl, whisk together the flour, instant yeast, sugar, and salt.

In a small saucepan, gently warm the unsalted butter with the milk until the butter has melted. Let it cool slightly.

In a medium bowl, whisk together the lukewarm water and one large egg. Stir in the milk and butter mixture.

Pour the liquid mixture into the dry ingredients and stir until a sticky, shaggy dough forms. If the dough is too sticky to handle, add 1 tablespoon of flour at a time until it becomes less tacky. Cover the bowl with plastic wrap and let it rise in a warm place for 2 to 4 hours, or until it has doubled in size.

Turn the dough onto a lightly floured surface, divide it into 8 equal pieces, and shape each into a round bun. Place them about 2 to 3 inches apart on a parchment-lined baking sheet. Cover loosely with a clean towel and let rise for about 1 hour until puffy.

Preheat your oven to 400°F. In a small bowl, whisk together the remaining egg and 1 tablespoon of water to make an egg wash. Brush the tops of the buns with the egg wash and bake for 15 minutes or until they are golden brown. Remove them from the oven and let them cool on a wire rack before serving.