1.Grease a 9×13-inch baking pan; set aside. Heat up 1/3 cup of butter with 1/3 cup of milk in the microwave until the butter is melted and mixture is lukewarm. Pour mixture into a bowl of a stand mixer.
2.Add 2 tablespoons honey, yeast, and warm water. Stir to combine. Add 2 cups flour and 1/2 teaspoon salt to yeast mixture. Using a dough hook, mix on low speed for about 1 minute.
3.With the mixer still going, add remaining flour, 1/2 cup at a time. Mix about 1 ½ minutes, or until dough starts to clean the sides of the bowl. Knead on low speed for about 2 more minutes, or until dough is smooth and elastic – the dough will still be slightly sticky to the touch.
4.Place dough in a greased bowl, turning it to grease the top. Cover the dough with a clean, dry dish towel. Let it rise in a warm place, free from draft, for about 10 minutes.
5.Meanwhile, in a small bowl mix brown sugar, sugar, cinnamon, 1/4 cup softened butter. Set aside. Turn the dough onto a floured surface, and roll into a 12×16-inch rectangle. Spread filling over dough.
6.Roll dough up lengthwise, pinching seam to seal. Cut into 12 pieces, and place in prepared baking dish. Cover the baking dish with the dish towel, and let the dough rise for about 15 minutes. Meanwhile, preheat oven to 350 F.
7.Bake at 20-30 minutes, or until the rolls are golden brown and baked through. Let the honey buns cool in pan for 5 minutes. Meanwhile, whisk 1 tablespoon melted butter, powered sugar, 1 tablespoon honey, and pinch of salt. Icing will be thick.
8.Spread icing over warm buns. Let buns sit 10-15 minutes before serving.