Chef John's Milk Bread
Ingredients:
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¼ cup bread flour
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¼ cup water
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⅓ cup whole milk
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2 ½ cups bread flour
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1 ¼ teaspoons kosher salt
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¼ cup white sugar
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1 (.25 ounce) package instant dry yeast
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½ cup warm whole milk (110 degrees F (43 degrees C))
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1 large egg, beaten
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4 tablespoons unsalted butter, at room temperature, cubed
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1 tablespoon milk, or as needed