Mix the melted butter and honey until smooth and set it aside (or refrigerate) until ready to be used.
Combine the lukewarm water and yeast in a large bowl or stand mixer and let it sit until it dissolves and starts to bubble, about 5 minutes.
Meanwhile, in a microwave-safe medium bowl, combine the milk, butter, and salt. Microwave for about a minute and stir until everything is melted. You can also do this in a small saucepan on the stovetop. Let it cool to 105-115℉ (41-46℃) to prevent it from killing the yeast and curdling the egg.
Lightly whisk the egg and honey into the milk, then dump everything into the yeast mixture.
Mix on medium-high speed using a dough hook, if using a stand mixer, until all the ingredients are combined. Or mix by hand with a large mixing spoon until well combined.
Mix in the whole wheat flour and a cup of all-purpose flour. Continue mixing the dough, adding more flour as needed, until the dough forms a soft dough and doesn’t stick to the sides of the bowl. The dough should be lightly sticky rather than stiff.
Turn the dough out onto a lightly floured surface and knead it for 8-10 minutes. The more you knead it, the better it rises because the gluten develops. The dough will be smooth and elastic.
Place it in a large greased bowl. Cover it loosely with a clean cloth and let it rise in a warm, draft-free place for about 45 minutes to 1 hour, or until doubled in size.
Punch the dough down, divide it into 12 equal balls (24 for smaller rolls), and shape them into balls. Place them in a greased 12-inch cake pan.
At this point, the dough may be shaped into a greased loaf pan and placed seam side down in the pan, or divided into 12 equal pieces. Shape them into balls. Place in a greased 12-inch round pan.
Brush with the honey butter mixture, cover lightly, and let them rise for 30-60 minutes, or until doubled in size.
Preheat the oven to 350℉ (180℃). Bake the rolls for 20-25 minutes or until beautifully browned and they sound hollow when tapped.
Remove them from the baking pan and let them cool for 20-30 minutes. Enjoy with more honey butter.