Melt 1/4 cup butter and set aside about 1 tablespoon to brush on the baked rolls; use the remaining melted butter for the dough and to grease the bowl.
Place 1/2 cup whole milk in a small saucepan and bring just to a boil. Immediately remove from heat and stir in 1/2 cup granulated sugar, the remaining melted butter, and 1 1/2 teaspoons salt. Let the mixture cool until lukewarm (not hotter than 115°F / 46°C).
In a heatproof measuring cup, combine 1/2 cup warm water (105°–115°F) and 2 packages active dry yeast. Stir briefly and let sit 5–10 minutes, until foamy.
In the bowl of a stand mixer (or a large mixing bowl), pour in the foamy yeast mixture, add the cooled milk mixture and 2 lightly beaten eggs. Add 2 cups of the all-purpose flour and beat with the paddle attachment just until combined.
Gradually beat in the remaining 2 1/2 cups flour until incorporated. The dough should come together into a soft, slightly sticky ball; you may need a little more flour—add by tablespoons if necessary.
Knead the dough: with a dough hook on speed 2, knead about 5–7 minutes until smooth and elastic. Or turn onto a lightly floured surface and knead by hand 8–10 minutes (you can do about 5 minutes on the hook and finish by hand if preferred).
Lightly grease a medium bowl with a little melted butter, place the dough in the bowl, and turn to coat all sides. Cover with a kitchen towel or plastic wrap and set in a warm place to rise until doubled, about 1 hour.
Punch the dough down. Divide into balls of about 2 to 3 tablespoons each and place them in a greased 9×13-inch pan (about 5 rows of 4 rolls). Cover and let rise in a warm place until nearly doubled, about 1 hour.
About 10 minutes before the end of the second rise, preheat the oven to 375°F.
Bake the rolls 20–25 minutes, or until golden brown on top. Immediately brush the tops with the reserved tablespoon of melted butter and serve.