In the bowl of a stand mixer fitted with the dough hook attachment, combine whole wheat flour, instant yeast, warm water, sugar, salt, cubed butter, and one egg. When doing the straight dough mixing method and adding everything at once, add the flour first — it incorporates faster and easier without yeast or other ingredients getting stuck on the bottom of the bowl.

Mix on low speed until all ingredients are incorporated with only a few butter pieces remaining. Room temperature butter will incorporate faster, but it isn't necessary.

Increase the mixer to medium speed and knead for 4 minutes, or until the dough pulls a rough windowpane. Whole wheat dough won't develop the same smooth windowpane as white flour dough.

Scrape dough into a large bowl sprayed with non-stick cooking spray or lightly coated with neutral oil. Cover and proof in a warm spot for 45 minutes to 1 hour, until visibly doubled and airy to the touch.

Spray a 9x13-inch baking dish with non-stick cooking spray or grease with softened butter.

Turn the dough out onto a lightly floured work surface and divide into 12 equal pieces (approximately 80g each).

Shape each piece into a roll by folding all the edges into the center, flipping seam-side down, and rolling quickly between your thumb and other fingers. Use a push-pull motion to create surface tension and form a ball. Work quickly so the dough doesn't stick excessively to your hands or counter. The center hole will not fully disappear with whole wheat dough when shaping, unlike with white flour rolls — this is normal!

Place shaped rolls in the baking dish and repeat with remaining dough. Cover with plastic wrap and proof in a warm spot for 1 hour, until tripled in size and light and airy when gently touched. This step is crucial for light, fluffy whole wheat rolls! Meanwhile, preheat the oven to 350°F conventional (no fan).

Before baking, gently brush the tops with beaten egg (egg wash). Be gentle and don't let it run down the sides or pool, as this will give the tops a shiny finish but prevent proper browning on the sides.

Bake rolls in preheated oven for 25-28 minutes until golden brown. You can also test with an instant-read thermometer — this enriched dough should read 185°F.

Serve slightly warm or at room temperature. Brush with melted butter if desired.