Place the grated cheese in a small bowl and cover to prevent drying then set aside. Combine 1 1/2 cups unsifted flour, yeast, sugar, and salt in a large mixer bowl, blending thoroughly. Measure 3/4 c of milk, water, and butter into a saucepan and heat until the liquids are warm, 115 to 120 degrees F. . Gradually add the liquids to the dry ingredients in the mixer bowl, beating for 2 minutes at medium speed of the electric mixer, scraping the bowl occasionally. Add and beat in 1 cup of unsifted flour at high speed. Beat for 2 minutes, scraping the bowl occasionally. Mix in enough additional flour (1/2 to 1 cup unsifted) to make a soft dough. (Dough will be slightly sticky. ) Put the dough into a greased deep bowl. Cover with waxed paper and a clean towel and let stand in a warm place until the dough has doubled, 45 to 60 minutes. Generously grease several baking sheets. Melt the butter and set aside. Punch the dough down wit a fist and turn the dough out onto a lightly floured surface. Divide the dough into two equal portions. Set one portion aside. Roll the dough into a rectangle 16 X 8-inches. Brush with about one-half of the melted butter. Sprinkle with about one half of the grated cheddar cheese. Cut crosswise into 8 equal portions. Cut into halves lengthwise. Fold each strip into thirds, lapping each side portion over the center third. Place the rolls on a baking sheet. Repeat for the other half of the dough. Beat the egg yolk with the tbs of milk, slightly. Brush the tops of the rolls with the egg yolk mixture. Let rise until doubled, about 30 minutes. Bake at 425 degrees F. for about 8 minutes or until rolls are golden brown. Serve rolls hot.