Whisk Tangzhong Base:

In a small saucepan, combine the water and bread flour, whisking until no lumps remain. Pour in the remaining milk and whisk again.

Cook Tangzhong:

Set the saucepan over medium heat and whisk continuously until the mixture thickens to a pudding-like consistency.

Cool Tangzhong:

Transfer the mixture to a bowl, press plastic wrap directly onto the surface to prevent a skin, and let it cool to room temperature.

Activate Yeast:

In the mixing bowl, stir together the lukewarm milk and about 1 tsp of sugar until dissolved. Sprinkle the yeast over the milk and stir gently; let stand for 10-20 minutes until bubbly and frothy.

Mix Dough Ingredients:

After the yeast activates, add the remaining sugar, dry milk powder, tangzhong, egg, flour, and salt. Stir with a spatula until a shaggy dough forms.

Knead Dough Initially:

Using the dough hook on low speed (2-3), knead for 5 minutes. The dough will be very sticky but will come together.

Incorporate Butter:

Add the softened butter in 3-4 portions, mixing for about 20 seconds between each addition. Scrape down the bowl as needed.

Knead Until Smooth:

Once the butter is fully incorporated, increase the speed to medium (4-5) and knead for 5-7 minutes, scraping the bowl once or twice, until the dough is smooth, silky, and pulls away from the sides.

First Rise:

Turn the dough out onto a lightly floured surface, shape into a ball, and return it to the bowl. Cover with plastic wrap and let rise in a warm spot until doubled, about 1-2 hours.

Optional Refrigeration:

Optional: Refrigerate the dough for a couple of hours for easier handling, or slow-proof in the fridge for about 12 hours.

Prepare Pans:

Butter a 4.5x8.5-inch loaf pan and dust with flour. For dinner rolls, butter a 9- or 8-inch square cake pan and dust with flour. Set aside.

Degas Dough:

Turn the proofed dough onto a lightly floured surface and press to release all gas.

Shape Loaf Rolls:

For the loaf: Weigh the dough and divide into 4 equal portions. Roll each into a 6-inch square. Fold opposite corners toward the center, then roll up tightly from the pointed end, pinching the seam to seal. Place seam-side down in the loaf pan (4 rolls per pan). Cover with plastic wrap and let proof in a warm place until doubled and just below the pan rim, about 1-2 hours.

Shape Dinner Rolls:

For rolls: Divide the dough into 9 equal pieces. Roll each into a smooth ball and place in the square pan, spacing ¼-½ inch apart. Cover with plastic wrap and let proof in a warm place until doubled, about 1-2 hours.

Bake Bread:

Preheat oven to 350°F (175°C). Brush the proofed bread with milk wash (matte crust) or egg wash (glossy crust). Bake loaf for 30-35 minutes, rolls for 20-30 minutes. If browning too quickly, tent with foil. Internal temperature should reach 190-205°F (88-96°C).

Cool Bread:

Remove from oven and let cool in the pan for a few minutes, then turn out onto a wire rack to cool completely. Bread is easier to slice at room temperature but can be enjoyed warm.