Preheat the oven to 350?F.
Combine the milk, 1/4 cup of the butter, 1/4 cup of the sugar, and the salt in a small saucepan over medium heat. Cook, stirring frequently, until the butter is melted and the sugar has dissolved, 2 to 3 minutes. Remove from the heat and set aside to cool to lukewarm before proceeding.
Combine the yeast and the remaining 1 teaspoon of sugar in 1/4 cup of warm water (110?F). Set aside until foamy, about 10 minutes. Add the milk mixture to the yeast mixture and stir to combine. Add the eggs and stir to combine. Using a wooden spoon, stir the flour and dry milk, 1 cup at a time, into the dough. The dough will be quite stiff and somewhat sticky. If the dough gets too stiff to stir with a spoon, use your hands to mix.
Lightly grease the inside of a large bowl with 1 teaspoon butter. Transfer the dough to the bowl and turn to coat. Lightly grease a piece of wax paper or plastic wrap with 1 teaspoon of butter and use it to cover the bowl. Set aside in a warm place until the dough is doubled in size, at least 3 hours. Melt the remaining 1/4 cup of butter and set aside.
After the dough has doubled, turn it out onto a lightly floured surface and knead until smooth and elastic, 2 to 3 minutes. Using a lightly floured rolling pin, roll to a 1/2-inch thickness. Using a sharp knife or dough cutter, cut the dough into 12 equal pieces. Arrange the dough in a metal 9 x 13-inch pan. Brush the tops of the rolls with the melted butter. Cover the rolls with plastic wrap and set aside in a warm draft-free area until doubled in size, about 1 hour.
Bake the dinner rolls on the center rack of the oven until golden brown and puffed, about 18 to 20 minutes.
Makes 12 rolls