In the bowl of a stand mixer, combine warm milk, yeast, sugar, salt, butter, and eggs.
Add in 5 1/2 cups of flour. Using a dough hook, turn the mixer on to a low speed. Once the flour starts to incorporate into the dough, increase the speed to a medium range. Slowly add the remaining 1/2 cup of flour until the dough pulls away from the sides of the bowl. The dough mixture should be slightly sticky and soft.
Transfer the dough to a lightly greased mixing bowl. Cover with a towel and let rise 90 minutes.
Lightly grease a baking sheet. Punch down the dough and form into 24 rolls. Place on the greased baking sheet in six rows of four.
Cover and let rise 1 hour.
Preheat oven to 375 degrees. Bake the rolls for 12 to 14 minutes, until lightly browned.
Remove rolls from oven and brush with melted butter.
Serve the rolls warm, or to cool, let rest on the pan for 15 minutes before transferring to a wire cooling rack. Once cooled completely, store in a plastic bag.