Put the water in a large bowl, sprinkle with yeast and a pinch of the sugar. Let stand 5 minutes, or until foamy.
In a small bowl, stir together warm milk and eggs. Add to the yeast mixture along with 3 cups of the flour and the remaining sugar; mix until well blended and sticky.
Add the butter and remaining flour and stir or beat with the dough hook attachment of your stand mixer until you have a soft, sticky dough; knead for about 8 minutes, until smooth and elastic.
Place back in the bowl, cover and let rise for an hour, or until doubled.
Divide the dough in half and on a lightly floured surface, roll into a rectangle that’s about 10×15-inches and about 1/4-inch thick. Brush each piece with half the melted butter and scatter with cheese.
Starting on a long side, roll the dough up into a log, and cut it crosswise into thirds.
Cut each piece into thirds to get 9 pieces.
Place cut-side-up into two 8 or 9-inch round pans, cover and let rise for another hour while you preheat the oven to 350˚F.
Bake for 30-40 minutes, until deep golden.