Make the Flour Paste: Combine the flour and milk in a microwave proof bowl. Nuke for about 30 seconds and whisk. If not thickened, nuke for another 10 seconds. Continue till thickened.
Combine the flour, sugar, salt and yeast in your mixer bowl. Whisk briefly to distribute evenly. Add the flour paste.
Heat the water, milk and butter to 120F and allow butter to melt. Once melted add to the flour unless temperature is over 130F. Too hot will kill the yeast. Start your mixer with the dough hook. When flour is worked in, add the egg. Mix for about 10 minutes. Add some flour to adjust if dough is too wet. Amount can vary based on size of the egg.
Turn out onto floured work table and fold several times. Place in a greased bowl and turn to coat top. Allow to rise till doubled
Preheat oven to 350F.
Divide the dough. This will make about 8 rolls @ about 115 grams each. I made mine 150 grams for 6 larger rolls because our hotdogs are longer than normal.
Pat each portion into a rectangle. Roll up and pinch to seal. Turn ends over, into the roll. Gently roll on table to meet desired size. Place on a parchment lined baking sheet. Remember that I purposefully made mine bigger as shown in the picture. Allow to rise for 30 - 45 minutes in a warm location.
Brush each roll with egg wash. Bake for about 20 minutes or till internal temperature of rolls is 190F. Allow to sit on baking tray for 10 minutes then remove to a wire rack to cool.