Add the warm milk and stir the yeast and brown sugar in a stand mixer bowl with a dough hook. Leave it to sit for 5 minutes until foamy.
Add whole wheat flour, all-purpose flour, egg, and salt. Turn the mixer on low and combine it into a shaggy dough. If mixing by hand, knead the dough into a shaggy dough ball, then transfer it to a workbench and knead on there.
Add a few cubes of cubed butter at a time and mix it on medium speed until the butter has been incorporated. At this point, the dough will be very sticky.
Keep the mixer on medium speed and keep mixing for around 6 minutes, or by hand for 8-10 minutes until the dough gathers strength and starts to pull away from the sides of the bowl. The dough will be soft and sticky, but it should be strong and hang off the dough hook in one cohesive mound.
Pull the dough from the mixer bowl and shape it into a ball. Lightly grease a clean, large bowl and place the dough ball in the bowl. Cover with a clean tea towel or plate and let it stand in a warm spot until the dough has about doubled in size, for about 1 hour, depending on room temperature.
Gently deflate the dough by punching it down, remove it from the bowl, and place it on a lightly floured bench. Roll the dough into a 15-inch length and cut it into 15 1-inch pieces. Shape each piece into a tight ball.
Grease or line a 9×13 baking dish with parchment paper. Place the dough balls into the prepared baking tray. Cover the tray with a clean tea towel and let it stand in a warm spot until the dough balls have doubled, around 40-60 minutes, depending on room temperature.
While the dough is rising, preheat the oven to 375°F/190°C. Brush the tops of the dough balls with milk.
Bake the rolls in the oven for approximately 22-25 minutes until deep golden brown. If they are browning too fast, loosely place a sheet of aluminum foil over the top of the rolls.
Brush the warm baked rolls with melted butter for extra soft tops. The bread rolls are best served warm.
On the following days, the rolls can be reheated in the microwave for 10 seconds.