In stand mixer bowl, add 1 cup flour, yeast, sugar, salt, Italian seasoning and Parmesan; mix well. In saucepan or microwave-safe dish, heat milk, water and oil to 120-130°F. Add to flour mixture. Using a paddle attachment, blend on low speed for 30 seconds; beat on medium speed for 3 minutes. Add egg; beat 1 minute.
Switch to dough hook attachment. Gradually stir in enough remaining flour to make a firm dough. Knead until smooth and elastic, about 5 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.)
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise in warm place for 15 minutes.
Prepare pan: Grease a 9×13-inch cake pan; set aside.
Divide dough into 4 equal parts. Divide each part into 4 equal pieces. Shape each piece into a smooth ball. Add topping: Dip tops into melted butter, then in shredded Parmesan cheese. Place in prepared pan. Cover; let rise until indentation remains after touching, about 20 minutes.
Preheat oven to 375°F.
Bake for 20 to 25 minutes until golden brown. Remove from pans; serve warm or cool on wire rack.