In a large liquid measuring cup, add ½ cup milk and warm until around 110 degrees F. Add the yeast and let sit for about 10 minutes.
Meanwhile, in a small saucepan add remaining ¼ cup milk, water, and ¼ cup bread flour and heat over medium heat stirring frequently until thickened. (Keep a close eye on it because it will thicken quickly, and you do not want it to burn). Remove from heat and set aside once thickened.
In the bowl of a stand mixer fitted with the dough hook attachment, add the remaining 2 ½ cups bread flour, salt, and sugar. With the mixer on low, slowly add the milk/yeast mixture until you get a crumbly consistency.
Next, add in the thickened flour mixture, 1 egg plus 1 yolk, and 4 tablespoons softened butter. Mix on high speed for about 8 minutes. The dough should only be slightly sticky.
Turn the dough out onto a lightly floured surface and form into a ball. Place in a lightly greased large bowl and cover with plastic wrap and set in a warm area for about 1 hour or until doubled in size.
Once the dough is doubled, punch down the center of the dough. Section the dough into 9 equal pieces (I used a food scale and weighed the entire ball of dough and then divided that weight by 9, mine was al ittle over 3 oz. per roll. You can also do them smaller if you wish)
Roll each piece of dough into a ball and place into a lightly greased 9X9 baking dish. Cover with plastic wrap and allow to sit in a warm place for 30-45 minutes or until doubled in size.
Before the time is up, preheat your oven to 375 degrees F.
Brush your rolls with an egg wash and bake for 25 minutes or until the inside reaches around 190 degrees F. (If the rolls get too brown, tent with a little aluminum foil).
Once the rolls are finished baking immediately brush with melted butter and serve warm.