In a small saucepan, bring the milk to a scalding temperature (when a skin forms on top of the milk, about 170 degrees), stirring frequently. Remove immediately from heat after scalding.
Meanwhile, combine 1/3 cup butter, sugar, and salt in a medium bowl. Pour scalded milk over the top and cool to 110 degrees to 115 degrees, stirring occasionally. Whisk in the eggs.
While the scalded milk mixture is cooling, add the yeast to the warm water (110 degrees) and let it "bloom" for 5 minutes (see note 2).
In an electric mixer fitted with the dough hook attachment, combine 4 1/2 cups flour, yeast, and water. With the motor running on low, slowly drizzle in the scalded milk mixture.
Increase the mixer speed to medium and mix until shiny and smooth, 6 to 10 minutes. If the dough is sticky after 3 minutes, add the remaining ½ cup flour, 1 tablespoon at a time, until the dough comes together. Using a small, microwave-safe dish, melt the remaining butter for 15 to 20 seconds.
Turn out the dough onto a heavily floured surface and shape into a ball. Place in a greased bowl and brush with 1 teaspoon melted butter. Cover the bowl with plastic wrap and let proof (rise) in a warm place (80 degrees to 85 degrees) until doubled in volume, about 1 1/2 to 2 hours (see note 3).