Mix the yeast, sugar and warm water. Let stand for five minutes to make sure the yeast is alive and starts frothing.
Add all of the whole wheat flour, about 1 ½ cups of the all-purpose, dried basil, olive oil, and salt and knead by hand or in a stand mixer. If the dough is too sticky, keep adding the remaining all-purpose flour until you get a smooth dough.
Place in an oiled bowl and turn the dough over once to coat it on all sides with oil. Cover with a kitchen napkin and place in a warm spot for about 1 ½ hours or until it doubles in volume.
After it has doubled, punch down the dough, and divide it into two. If you're making just one pizza, this would be a great time to wrap one half in plastic wrap and freeze it for future use. When you need it, just bring it out several hours before and let it sit at room temperature to thaw.
Shape the other half into a ball and let it sit around 10 minutes. Then, on a lightly floured surface, roll it into a round of about 14 inches (this isn't a thin crust but I still roll it fairly thin. Make sure your baking sheet is large enough to accommodate the rolled crust).
Push down around the center with your fingers so you get a slightly raised edge around the pizza.
Dust the baking sheet with cornmeal and place the crust on it. Brush the crust with olive oil and place in a 475-degree oven for about 8-10 minutes until lightly golden. The crust might bubble up in places, but don't worry. It adds to the great, rustic look.
Take the pizza out of the oven and spread the pesto sauce over it. Then layer the mushrooms and onions and finally the basil over it.
Place it back in the oven for another 3-4 minutes.
Cut into slices and dig in!