Dissolve the yeast in the warm water with the sugar. After 5 minutes stir in the salt, johnnycake meal, whole-wheat flour, and oil. Gradually add the white flour, stirring with a wooden spoon until a stiff dough has formed. Place the dough on a floured board, and knead it for several minutes, adding only enough additional flour to keep the dough from sticking. When the dough is smooth and shiny, transfer it to a bowl that has been brushed with olive oil. To prevent a skin from forming, brush the top of the dough with additional olive oil, cover the bowl with plastic wrap, and let rise in a warm place, away from drafts, until doubled in bulk, 1 1/2 to 2 hours. Punch down the dough and knead once more. Let the dough rise again for about 40 minutes. Punch down the dough. If it is sticky, knead in a bit more flour.
Divide into four balls. Cover the balls with plastic wrap and allow to rise at room temperature for about 45 minutes. While the dough is rising, prepare a hot charcoal fire, setting the grill rack 3 to 4 inches above the coals and set out topping ingredients. Place dough on a large inverted cookie sheet brushed with 1 to 2 tablespoons extra virgin olive oil. Turn the dough over to coat it with oil. With your hands, spread and flatten the pizza dough into a 10 to 12-inch free form circle, 1/8-inch thick. If you find the dough shrinking back into itself, allow the dough to rest for a few minutes, then continue to spread and flatten the dough. Do not make a lip, you may end up with a rectangle rather than a circle; the shape is unimportant. Take care not to stretch the dough so thin that small holes appear. If this happens, all is not lost. Rather then try to repair them, avoid them when adding toppings and drizzling with olive oil.
When the fire is hot, use your fingertips to lift the dough gently by the two corners closest to you, and drape it onto the grill. Catch the loose edge on the grill first and guide the remaining dough into place over the fire. Within a minute, the dough will puff slightly, the underside will stiffen, and grill marks will appear.
Using tongs, immediately flip the crust over onto the coolest part of the grill. Quickly brush the grilled surface with 2 teaspoons of virgin olive oil. Spread 1/4 cup of Parmigiano-Reggiano cheese, 2 tablespoons Pecorino Romano cheese, 1/4 cup Fontina cheese over the entire surface of the pizza. Dollop with 6 tablespoons of the tomatoes and top with 1 tablespoon of the parsley. Drizzle the entire pizza with virgin olive oil.
After the toppings have been added, slide the pizza back toward the hot coals so about half of the pizza is directly over the heat. Rotate the pizza frequently so that different sections receive high heat checking the underside by lifting the edge with tongs to be sure it is not burning. The pizza is done when the top is bubbling and the cheese has melted. Garnish with basil and serve immediately. Continue stretching the dough balls and grilling pizzas using the above topping ingredients.