To make the tangzhong, combine the flour, milk, and water in a small saucepan. Cook over medium-low heat, whisking continuously, until thickened into a pudding-like paste. Remove from the heat and let it cool until it is no longer hot.
⅓ cup (45 grams) bread flour, ½ cup (120 milliliters) whole milk, ½ cup (120 milliliters) water
To make the dough, combine flour, brown sugar, yeast, and salt in the bowl of a stand mixer. Then add the cooled tangzhong, pineapple juice, egg, and vanilla and mix with the paddle attachment on low until the dough comes together.
5 cups (630 grams) bread flour, ½ cup (98 grams) brown sugar, 2 teaspoons instant yeast, 2 teaspoons (12 grams) salt, ¾ cup (180 milliliters) pineapple juice, 1 large egg, 1 teaspoon vanilla extract
Change to the dough hook and knead on medium speed. Add the softened butter a few cubes at a time, allowing each addition to incorporate before adding more.
4 tablespoons (57 grams) unsalted butter
Continue kneading until the dough is smooth and elastic and starts to pull away from the bowl. Try to resist the urge to add more flour unless the dough is very very sticky.
Shape the dough into a ball, put it in a lightly greased bowl, and cover. Let rise in a warm place until doubled in size, about 1.5 to 2 hours.
Grease a 9 by 13-inch baking pan. Punch the dough down and divide into 15 equal pieces
To shape, flatten each piece slightly and form a ball by bringing the edges into the center little by little, working your way around the piece. Pinch the seam together, flip it over, turn it seam-side down, and roll the ball against the counter with a cupped palm to form a smooth, taut ball. Arrange the dough balls in the prepared baking pan in five rows of three.
Cover the pan and let your Hawaiian rolls rise until the rolls are noticeably puffy, about 1 to 2 hours. When gently pressed with a fingertip, the indentation should remain rather than immediately springing back.
Thirty minutes before baking, heat the oven to 350°F. Brush the tops of the rolls gently with the beaten egg and bake for 20 to 25 minutes, until the rolls are golden brown and the internal temperature of a center roll reaches about 190°F. If the tops brown too quickly, tent the pan loosely with aluminum foil.
1 large egg