Heat the milk in a small saucepan until very warm to the touch. Pour ¼ cup milk into a measuring cup and sprinkle the yeast over the surface. Let it sit a minute, then stir well until yeast is dissolved. Let stand for 4-5 minutes to activate the yeast.
Place the saucepan of remaining milk over low heat and add the butter, sugar and salt, whisking until thoroughly dissolved and mixture is smooth. Remove from heat and set aside with the yeast.
Measure 3 cups flour into a large bowl and whisk vigorously to remove any lumps. When yeast has sat 4-5 minutes, combine it with the warm milk mixture, then whisk in the egg and yolk until smooth. Pour over the flour and stir firmly until a dough forms. Dust a clean work surface with a generous ¼ cup flour and turn out the dough. Using floured hands, knead the dough until smooth and elastic, adding small amounts of flour if needed to form a soft dough. Knead about 4-5 minutes until smooth and elastic, and until dough holds a soft ball shape. (Or place in standing mixer with the dough hook and mix on medium speed 6-7 minutes, adding flour gradually until no longer sticking to sides of bowl.)
Lightly oil the inside of a large, clean bowl and place dough inside. Pull sides of dough in slightly toward the center and flip the dough over so that the oiled side is up. Cover bowl with plastic wrap and place in a warm place at room temperature until doubled in size, about 1 ½ hours.
When ready to shape rolls, melt the 5 TB butter in a small saucepan and keep warm over low heat. Remove plastic wrap from bowl and punch down dough. Pull sides of dough slightly in toward center, then turn dough out onto a well-floured surface. Roll dough with a rolling pin to about ½” thick. Using a round cutter with a diameter of 2”-2 ½”, dipped in flour before each cut, cut rounds of dough as close together as possible.
Pour the warm butter into the bottom of a 9x13-inch metal or glass baking pan. (Set aside the saucepan with the butter that remains in it.) Picking up 1 piece of dough at a time, dip both sides into the butter, then fold in half and place into the pan. Repeat with the remaining rounds, placing the rolls very slightly apart. From the scraps of dough remaining, press together similar-sized rolls and coat in butter before placing in pan. Pan should be filled, with rolls evenly spaced.
Cover pan loosely with plastic and let rise in a warm place at room temperature about 45 minutes, until puffy, though not fully doubled. Meanwhile, preheat oven to 350`F. Gently warm the saucepan with remaining butter and remove from heat. If desired, whisk in the egg white. Before rolls go into the oven, brush the tops with the butter/egg mixture. The egg will give the tops a shiny surface. Otherwise, brush with just the butter. Lightly sprinkle the tops with salt if using. Bake rolls for 25-30 minutes, or until golden, puffed and springy. Serve warm.