In a medium bowl, whisk together the lukewarm milk, molasses, melted butter, and the egg until the ingredients are well combined.
In a large mixing bowl (or the bowl of a stand mixer), combine the whole wheat flour, all-purpose flour, salt, and yeast. Add the wet mixture, stirring until a shaggy dough forms.
If using a stand mixer, with the dough hook, knead on low speed for about 6 minutes. If kneading by hand, work the dough on a lightly floured surface for about 12 minutes. The dough should feel smooth and elastic by the end.
Cover the dough with plastic wrap or a damp towel, and let it rise in a warm place for about 1 hour and a half, or until it has doubled in size.
After the first rise, divide it into portions based on your preferred roll size. For large rolls, divide into 8-9 portions. For medium rolls, divide into 12-14 portions. For small rolls, divide into 18-20 portions.
Shape each piece into a ball by stretching the dough over itself and pinching the edges together. Place the rolls in a lined baking dish.
Cover the shaped rolls and let them rise for 40 minutes to 1 hour.
Preheat your oven to 350°F, positioning a rack in the lower-middle of the oven.
In a small bowl, mix the remaining melted butter with 1 teaspoon of molasses. Brush this mixture lightly over the tops of the rolls. If you prefer, save half to brush on after baking for extra flavor and shine.
Bake for 25 to 30 minutes, or until the rolls are golden brown on top.
Let the rolls cool for about 10 minutes before serving.