In a small bowl, soften yeast in the 2/3 cup warm water.
In a large bowl, beat butter with electric mixer for 30 seconds. Add sugar and salt; beat until fluffy. Add 1 cup of the flour, the 6 eggs, and the yeast mixture. Beat on low speed until combined. Beat on medium speed for 2 minutes. Using a spoon, stir in Parmesan cheese and remaining flour until smooth. Cover; refrigerate at least 4 hours or overnight.
Punch dough down. Turn out onto a floured surface. Divide into 6 portions; divide each into 4 pieces. With floured hands, form each piece into a ball. Place in 24 well-greased individual (about 4-inch) brioche pans, 2-3/4-inch fluted muffin pans, or 3-1/2-inch muffin pans.
Beat together the 1 egg and the 1 tablespoon water. Brush tops with the egg mixture. Cover and let rise in a warm place until nearly doubled (45 to 60 minutes).
Bake in a 375 degree F oven about 15 minutes or until golden, brushing with egg mixture after 7 minutes. Remove; cool slightly. Makes 24 rolls.
