Mix 2 1/4 cups all-purpose flour, whole wheat flour, Parmesan cheese, Italian seasoning, salt, and yeast. Stir in water and oil and mix until smooth. Add in mozzarella and red pepper and add in enough of the rest of the flour to form a soft dough.
Knead dough until elastic, about 5 minutes. Lay dough in a greased bowl and let rise, covered in a warm place 30 minutes, dough should be doubled in size. Punch down dough.
Form dough into a loaf. Lay loaf in a greased 9 x 5 inch loaf pan. Let rest, covered, 30 minutes or until doubled in size.
Slice slits into top of dough. Whisk together egg whites and water and brush on top of bread. Sprinkle bread with Italian seasoning. Bake at 375 degrees F 35 to 40 minutes. Let cool before slicing.