Add water, yeast and sugar to a bowl and let bloom for 5 minutes.
To the bowl of a stand mixer fitted with the dough hook, add the sugar, yeast, water, milk, oil, salt, and 3 cups of the flour. Stir with a wooden spoon until a rough dough comes together. Start kneading with the dough hook. If necessary, you can add some (or all) of the remaining cup of flour, 1 tablespoon at a time, until the dough pulls away from the sides of the bowl and sticks only to the bottom. The dough should be tacky, but not sticky. (I didn’t need to use any additional flour beyond the 3 cups, but this will vary depending on the conditions in your kitchen.) Knead the dough on low speed for about 5 minutes, or until it is smooth and elastic.
Spray a large bowl with nonstick cooking spray. Shape the dough into a ball and transfer to the prepared bowl. Cover and let rise until doubled in size, about 1 hour.
Line a baking sheet with parchment paper. Turn the dough out onto your work surface and divide it into 10 equal pieces.
Working with one piece of dough at a time, shape it into a ball and then roll the ball into a cylinder measuring about 4 1/2-inches in length. Transfer to the prepared baking sheet and flatten the cylinder slightly. Repeat with the remaining pieces of dough, placing them only about 1/2-inch apart on the baking sheet. Once they’re all shaped, cover the buns with a towel and let rise until almost doubled in size, about 45 minutes.
Meanwhile, preheat oven to 400 F. Brush the buns with the egg wash and then bake for about 12 minutes, or until the tops are golden brown and an instant-read thermometer reads 190 F. Remove the buns from the baking sheet to a wire rack and cool completely.