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Combine the flour, granulated sugar, yeast, and salt in the bowl of a stand mixer fitted with a dough hook attachment.
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Add the milk, butter, and half of the water. Mix until the dough comes together. Slowly add the remaining water to form a sticky dough.
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Mix on high speed until the dough is smooth and elastic, about 8 minutes.
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Place the dough in an oiled bowl and cover with plastic wrap. Let rest for 45 minutes or until doubled in size.
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Line a baking sheet with parchment paper. Set aside.
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Place the dough on a lightly floured surface and punch down the dough to release the excess air. Knead the dough for two minutes or until smooth.
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Divide the dough into six pieces and then roll them into a ball using the cup of your hand.
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Place the dough balls onto the prepared baking sheet. Cover with plastic wrap and let rest for 30 minutes. Preheat the oven to 425 F.
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Combine the egg and water to make an egg wash. Brush the egg wash over the dough balls.
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Bake the buns for 10 minutes or until golden brown. Let the buns cool on a wire rack.