Place the milk in a small saucepan and bring to a simmer over medium-high heat. Remove from the heat, stir in the butter and sugar and let cool. Dissolve the yeast in the warm water and let sit until foamy, about 5 minutes.
Combine the milk mixture, eggs, yeast mixture, salt and 3 cups flour in a mixer fitted with the dough hook and mix until smooth. Add the remaining 3 cups flour, 1/2 cup at a time, and mix until a smooth ball forms.
Knead the dough by hand on a floured surface for about 5 minutes. Butter a large bowl, add the dough, cover and let rise in a warm place until doubled in size, 60 to 70 minutes.
On a floured surface, punch down the dough and cut into 24 two-inch pieces; roll into balls. Place on 2 parchment-lined baking sheets. Cover again and let rise until doubled, 30 to 40 minutes.
Arrange racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Bake the rolls until golden brown, about 20 minutes, rotating the pans halfway through. Remove from the oven and brush with melted butter before serving.