In the bowl of a stand mixer, combine the sugar, yeast, and whole wheat flour and all but 1 cup of the bread flour. Stir well.
Add the water, milk, oregano and olive oil. Gradually add remaining bread flour until you have a dough that is not too sticky to knead. Attach dough hook to mixer and let the machine knead for about 6 minutes or until smooth an elastic. If kneading by hand, it might take up to 10 minutes.
Transfer to an oiled bowl and let rise, covered, in a warm place (85°) free of drafts. With the quick yeast this should take about 45 minutes.
Punch dough down. Cover and let rest 5 minutes. Turn dough out onto a lightly floured surface. Arrange olives over dough; knead gently 4 to 5 times or until olives are incorporated into dough. Cover and let rest 10 minutes.
Preheat oven to 375°. Punch dough down and divide into 16 portions. Roll each portion into a 2-inch ball and place on baking sheets (or in 9x13 inch pans) coated with cooking spray. Cover and let rise 30 minutes or until doubled in size.
Cut a ¼-inch-deep "X" in the top of each roll. Combine water and egg white; brush over rolls. Bake at 375° for 18 minutes or until golden brown. Remove from oven; immediately sprinkle with cheese. Serve warm.