Place the warm milk in the bowl of a stand mixer. (Alternatively you can mix the dough in a bowl and then knead it by hand.)
Sprinkle the yeast over the milk and add the 1 teaspoon of sugar. Stir gently. Let the mixture sit for 5-10 minutes. The yeast should start to foam. If it does not, start over with new yeast.
Add they honey, salt and egg and mix gently to combine.
Add the 3 1/4 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. You may need to add up to 1/4 cup more flour, 1 tablespoon at a time. The dough will stick to your finger a little bit when you touch it but not be so sticky that it won't be workable. Avoid adding too much flour as that will create tough, dense rolls.
Transfer the dough to a lightly oiled bowl and turn once to coat all sides of the dough ball with oil. Cover and let rise in a warm place for 1 hour, or until doubled in size.
Grease an 8x8 or 9x9-inch square baking dish with butter. I like to use a glass Pyrex dish so that I can see the sides of the rolls to check when they are done.
Transfer the dough to a lightly floured work surface and knead a few times by hand. Cut the dough into 12 equal sized pieces. Shape each piece into a ball and place in the greased baking dish.
Cover the rolls in the dish and let rise for 20-30 minutes.
Meanwhile, preheat the oven to 400 degrees F.
Uncover and bake the rolls for 15-20 minutes, until the sides are very light brown and the rolls are baked through. Check on the rolls after 11-12 minutes and loosely cover the dish with foil to prevent the tops from over-browning. My rolls took 18 minutes total to bake.
Remove rolls from the oven. Whisk together the 1 tablespoon of melted butter and 2 teaspoons of honey. Brush the honey butter over the tops of the rolls. Serve warm.
Rolls will keep at room temperature, tightly wrapped, for 2 days, or in the freezer, wrapped airtight, for up to 2 months. It is best to freeze rolls before brushing with honey butter. Defrost and warm rolls in a 350 degree oven, covering as necessary. Brush warm rolls with honey butter.