In the bowl of a stand mixer fitted with the dough hook attachment or large mixing bowl, dissolve the yeast and sugar in the warm water. Let it sit for 15 minutes, until foamy.
While the yeast is working, whisk together the melted butter, honey, egg, milk, and salt. Make sure everything is dissolved and thoroughly combined.
Once the yeast mixture puffs up and looks really foamy, add all the whole wheat flour. With the mixer on the lowest setting, slowly add the whisked wet ingredients. Stir for 2-3 minutes, until the dough comes together. (If mixing by hand, just mix with a stiff silicone spatula and knead it until there are no dry flour bits.) Cover the dough ball with a clean kitchen towel, and proof in a warm place for 1 to 1½ hours, or until it doubles in size.
While the dough is proofing, brush a 13x9-inch (33x23 cm) casserole dish with oil and set aside.
Once the dough has doubled in size, punch it down, and hand-knead on a lightly floured surface for 1-2 minutes. Then divide the dough into 15 equal pieces and cover them with your kitchen towel.
Now take a dough ball, and fold the edges inward about 8 times as you rotate the dough to make a dense package. Then pick it up and close the open ends to form a smooth ball. Place it in your baking pan, and repeat until all 15 dough balls are in the pan. Cover the buns once again with your clean kitchen towel, and let them proof on the counter for another hour, until the buns almost double in size.
Position a rack in the center of the oven and preheat to 350°F/175°C. Bake the buns for 20-22 minutes until the tops are golden brown.
Once baked, immediately brush the tops of the buns with sugar water to get that nice shiny look. Carefully remove the buns from the baking pan and let them cool on a rack until warm (not piping hot). Serve immediately!