Preheat oven to 350°F. Mix one cup of very warm milk (should feel slightly warm when a drop is placed on the inside of your wrist) with 1 cup sugar, 1 stick of unsalted butter and 1 teaspoon of salt in the bowl of an electric mixer with the paddle attachment
Sprinkle the yeast over the remaining warm milk (in a separate bowl or cup). Add 1 teaspoon sugar. Stir until dissolved. Set aside until yeast is bubbly, about 10 minutes. Meanwhile, beat eggs until foamy; add eggs and yeast mixture to milk/butter/sugar mixture. Beat in 5 cups of flour, gradually adding 1 cup at a time until a soft dough forms. Change to dough hook.
Knead 5-8 minutes with machine. Turn out onto a clean work surface and continue to knead by hand 1-2 minutes, until dough is smooth. Butter or oil the inside of a non-metallic bowl and place dough inside, turning over once so that dough is coated and won't dry out. Cover with a clean damp towel or plastic wrap, then place in a warm place, free from any drafts.
Allow to rise for about 90 minutes, or until doubled in bulk. Punch dough down. Turn out, again, onto a clean work surface. Divide dough evenly into 12 round pieces and form buns 3/4 inch thick. Arrange on 2 buttered cookie sheets.
Make a hollow depression in the center of each with the back of a tablespoon. Press an uncooked egg, gently into the depression. Roll remaining dough into two long thin log shapes and cut into 12 pieces (each). Form each into a thin strip, 6 inches in length. Place two strips over each egg, in criss-cross fashion, being sure to tuck in the ends under the bun. Allow to rise again for about 45-90 minutes or until doubled in bulk (depends on the temperature in the room).
Beat an egg with 1 teaspoon of water to make a wash; use mixture to gently brush tops of risen buns. Cover loosely with aluminum foil and bake for 15 minutes or til golden. Remove to wire rack to cool. Brush lightly with melted butter.