Combine the whole milk, unsalted butter, salt, and 3 Tbsp. granulated sugar into a small saucepan over medium heat. Heat until butter melts, about 5 minutes. Stir to combine, and remove from heat.
Combine active dry yeast, 1 tsp. granulated sugar, and 1/3 cup warm water together into a glass measuring cup. Stir to combine. Allow yeast to bubble and activate, at least five minutes.
In the large bowl of a stand mixer, use the paddle attachment to combine all of the milk mixture with two cups of bread flour. Beat on low to combine, about two minutes.
Add the yeast mixture. Slowly work the speed up to high, and beat for three minutes.
Crack an egg into a small bowl, and use a fork to generously beat the egg. Add the egg to the dough mixture, and beat on a lower speed for about a minute to fully combine.
Remove the paddle attachment and switch to the dough hook attachment. Add all-purpose flour, 1/4 cup at a time, until fully mixed in. The dough should be tacky, but not completely wet.
Cover the bowl with a clean kitchen towel, and find the bowl a warm home for the next hour to let it rise.
After the dough has doubled in size, punch the dough down. Cover a clean surface with a sufficient amount of all-purpose flour, and divide the dough into fifteen equal pieces. Roll each piece into a small dough ball, and organize the dough balls into five rows of three inside a greased 9x13 baking dish. Cover the baking dish with the same kitchen towel you used earlier, and allow the dough balls to rise in a warm place for at least another 45 minutes, upwards of an hour.
Preheat the oven to 375 degrees Fahrenheit. Brush the tops of the rolls with egg wash, and then bake the rolls for 15-18 minutes, until the tops are a gorgeous, golden brown hue. Remove from the oven, and allow the rolls to rest for a bit as you paint melted butter over the tops of them (optional).
Serve warm, or keep in an airtight container for up to four days.