Add warm water, warm milk, 1/4 c. softened butter, honey, and eggs to a bowl of a stand mixer*. Add 2 1/2 c. flour and salt. Add yeast to the center.
Affix mixer with dough hook. Mix on speed 1 for 5-7 minutes or until dough ball forms and cleans the sides of the bowl. Note: add more flour, 1-2 Tbsp. at a time, if the dough is too sticky and doesn’t form a ball.
Do not remove dough from bowl.
Cover the dough with a clean dish towel and allow to rise 45 minutes.
Preheat oven to 350. Bake buns 20-25 minutes at 350 or until golden. Brush buns with remaining 2 Tbsp. melted butter.