Place the warm milk in the bowl of a stand mixer. (Alternatively you can mix the dough by hand in a large bowl and then knead it by hand.)
Add the granulated sugar and sprinkle the yeast over the milk. Whisk gently to combine. Let the mixture sit for 5-10 minutes, until the yeast starts to foam and/or bubble.
Add the honey, salt and egg and mix gently to combine.
Add the 3 ¾ cups of flour. Use the paddle attachment to mix on low speed until the flour is mostly incorporated. Add the butter cubes and continue mixing on low speed until the butter is incorporated.
Switch to the dough hook and knead the dough on low speed until it is smooth and slightly sticky to the touch, about 8 minutes. The dough is supposed to be slightly sticky, but if it is very sticky you can add a little more flour, 1 tablespoon at a time. Avoid adding too much flour, because this will make the rolls more dense instead of light and fluffy. The dough should stick to your finger a little bit when you touch it but not be so sticky that it won't be workable.
Transfer the dough to a large lightly oiled bowl and turn once to coat all sides of the dough ball with oil. Cover the bowl with a clean kitchen towel or plastic wrap and let rise in a warm place for 1 hour, or until doubled in size.
Grease a 13x9-inch rectangular baking dish with cooking spray or butter.
Transfer the dough to a lightly floured work surface and knead a few times by hand.
Cut the dough into 24 equal sized pieces. To do this, first cut the dough into 3 equal pieces. Then cut each of the 3 pieces into 8 pieces by first cutting it in half, then cut each half in half, and then cut each piece in half once more. You should end up with 24 roughly equal pieces of dough. Shape each piece of dough into a ball and place in the greased baking dish.
Cover the rolls in the dish and let rise for 30 minutes.
Meanwhile, preheat the oven to 400 degrees F.
Uncover and bake the rolls for 16-20 minutes, until the tops are golden brown and the rolls are baked through. (Mine usually take about 17 minutes to bake.)
Remove rolls from the oven. Whisk together the 1 ½ tablespoons of melted butter and 1 ½ tablespoons of honey. Brush the honey butter over the tops of the rolls. Serve warm.