How to Make It
In a large bowl, combine the flours, yeast and 1 teaspoon of salt. In a small bowl, whisk together the water, 1 tablespoon of the oil and the honey, then stir the liquid into the dry ingredients. Scrape the dough out onto a lightly floured work surface and knead until smooth. Shape the dough into a ball and transfer to a lightly oiled bowl. Cover with plastic wrap and let rise until doubled in bulk, about 30 minutes.
Preheat the oven to 500°. Punch down the dough and divide it in half. On a lightly floured surface, roll out 1 piece of dough to a 15-inch round, about 1/4 inch thick. Transfer to a large pizza pan or baking sheet. Repeat with the second piece of dough. Bake the dough for about 6 minutes, or until just dry and set.
Light a grill or leave the oven on. On a small baking sheet, toss the mushrooms with the remaining 2 tablespoons of oil and season with salt and pepper. Spread the mushrooms in an even layer. Set the baking sheet on the grill, cover with the lid and cook the mushrooms over a hot fire, or roast in the oven for about 3 minutes, or until softened.
Top the pizza crusts with the mushrooms, artichoke hearts, prosciutto, Fontina, mozzarella and Parmesan. Slide 1 pizza onto the grill and cook over a medium hot fire, or bake in the oven for about 5 minutes, rotating the pizza as necessary, until the cheese is melted and the bottom is browned and crisp. Using 2 large spatulas, return the pizza to the pizza pan, cut into wedges and serve at once while you cook the second pizza.