Mix all of the all purpose flour, whole wheat flour, water and instant yeast with a wooden spoon in a medium-sized bowl until well-blended. Cover the bowl, and leave it at cool room temperature (68°F-70°F) for 12 to 20 hours, until the biga is very bubbly.

Mix the biga and the remaining dough ingredients, except the cheese, using a mixer. Set on slow speed, for 2 to 4 minutes. Increase the speed to medium and mix for about 4 minutes; the dough should be soft and slightly sticky. Add additional water or flour if necessary. Mix in the cheese; don't worry if some pieces pop out. Allow the dough to rise, in a greased, covered bowl, for 1 to 2 hours, until it's very puffy. Note: You can also mix this dough in a bread machine set on the dough cycle; add the diced cheese several minutes before the end of the final kneading cycle.

Put the dough out onto a well-floured surface, and shape it into two long loaves, about 12 x 4 inches each.

Form into loaves by folding sides down as pictured. Pinch the back together to seal. Flip over and place the loaves, floured side up, onto parchment paper (if you plan to bake on an oven stone) or baking sheets. Cover the loaves with a greased plastic wrap, and allow them to rise for 45 minutes, or until puffy. Sprinkle with additional grated cheese.

Bake bread in a preheated 450°F oven for 22 to 26 minutes, until it's golden brown. Remove it from the oven, and cool on a rack.

Recipe Source: King Arthur Flour