Mix warm water with sugar in a large mixing bowl until the sugar has dissolved. Sprinkle yeast over the surface of the water, and let stand until the yeast begins to form a creamy layer on the water. Stir the yeast into the water. Stir in salt, and 1 cup of bread flour, and beat the flour in to form a loose batter. Stir in 1 more cup of flour, and mix in the mozzarella cheese, milk, and basil. Mix in the remaining 2 cups of flour in 1/2-cup additions.
Turn the dough out onto a well-floured work surface, and knead until the dough is thoroughly mixed, but still sticky, 5 to 8 minutes. Add flour into the dough as you knead, if necessary. Form the dough into a ball, and place into an oiled bowl. Turn the dough over in the bowl to oil the surface of the dough, and cover with a cloth. Allow to rise until double, about 1 hour.
With your fist, press firmly on the dough to deflate the large bubbles. Turn the dough out onto a well-floured work surface, and knead the dough just long enough to eliminate the remaining bubbles in the dough, about 1 minute. Cut the dough into 2 equal pieces, and form each half into a round ball. Spray a baking sheet with nonstick cooking spray, and place the balls onto the baking sheet. With a sharp knife, slash an X into the top of each loaf. Cover the loaves with a towel, and let rise until doubled, about 25 minutes.
Preheat oven to 425 degrees F (220 degrees C).
Bake the loaves until browned, about 25 minutes. To serve, cut each loaf into 8 wedges.