Position a rack in the lower third of the oven; preheat to 200 degrees . Grease a large glass or ceramic bowl. In a small pan, melt 4 tbsp. butter over medium heat. Add the milk and 1/2 cup water. Cook, stirring, until the mixture registers 125 degrees on an instant-read thermometer. Remove from the heat.
In the bowl of an electric mixer, combine 4 cups flour, the sugar, yeast and salt on low speed. Gradually blend in the warm milk mixture. Increase the speed to medium and mix for 30 seconds to blend. Mix in remaining 1/4 cup flour until the dough comes together in a sticky ball.
Attach the dough hook. Knead the dough on medium-high until smooth, 5 minutes. Transfer the dough to the prepared bowl (dont use the metal mixer bowl). Turn off the oven; let the dough rise in the oven for 10 minutes. Meanwhile, line a 10-inch springform pan or a 9-by-13-inch baking pan with parchment and grease.
Turn out the dough onto a floured work surface. Cut into quarters, then quarter each dough piece, to make 16 pieces. Quickly roll the dough pieces into balls by cupping your hands on top of them and rotating them against the work surface; nestle in the prepared pan. Cover with a kitchen towel; let rise in a warm spot for 10 minutes. Meanwhile, preheat the oven to 375 degrees .
Bake the rolls until golden-brown, about 25 minutes for the springform pan or 20 minutes for the baking pan. Melt the remaining 1 tbsp. butter; brush it on the tops of the hot rolls. Let the rolls cool for at least 20 minutes before serving.