Preheat the oven to 500 degrees F.
For the parsley pesto: Add the Parmigiano-Reggiano, pine nuts, basil, garlic, parsley and some salt and pepper to a food processor. With the motor running, slowly add about 1/2 cup olive oil and process until smooth. Pulse a few times to incorporate, adding more oil if the mixture is too thick. Taste and adjust the seasoning if needed.
For the mushrooms: Heat the canola oil in a large saute pan over medium-high heat until almost smoking. Add the mushrooms, garlic and shallots and saute until brown, about 5 minutes. Season with salt and pepper. Remove to a plate and reserve.
Preheat a grill pan over medium-high heat.
For the pizzas: Portion the dough into six golf-ball-size pieces. Dust a work surface with flour and roll each piece of dough into a rectangle (or whatever shape you like). Brush the tops of each piece with oil, sprinkle with salt and pepper and toss on the grill, oil-side down. Cook until the bottoms are an even golden brown, about 3 minutes. (The tops will also start bubbling up with air pockets.) Flip over and grill until golden brown on the other side and just firm, about a minute or so. Remove to a baking sheet.
Top the pizzas with the fontina, mushrooms, Parmigiano-Reggiano, dollops of goat cheese, crushed red pepper flakes and salt and pepper. Bake in the oven until the cheese is melted, 3 to 4 minutes. Top with the parsley pesto and garnish with a few more torn basil leaves.
In a small bowl, add 1/2 cup warm water (if the water is too hot the yeast will die and if the water is too cold the yeast will not activate). Add the dry yeast and sugar, stir to combine and let sit for 15 minutes. The top of the water will bubble and it will smell very yeasty; this is what you want.
In a large mixing bowl, combine the flour and salt. Make a well in the center of the flour and add the olive oil and the yeast/water mixture. Using a fork, gradually stir the flour into the yeast/water mix. Once most of the flour has combined, turn the dough out to a smooth, clean work surface dusted with flour and use your hands to knead the dough to a smooth consistency.
Lightly coat the mixing bowl with oil, return the dough to the bowl, cover with plastic wrap and set in a warm place until the dough has doubled in size, at least 1 hour.
When the dough is ready, portion it into six golf-ball-size pieces. (If you're not using the dough right away, wrap each ball individually in plastic and refrigerate until ready to use.) Dust a work surface with flour and roll each piece of dough into a rectangle (or whatever shape you like; I like irregular shapes). Top the pizzas as desired.
Pizza Dough adapted from "Cook Like a Rock Star" by Anne Burrell. Copyright Clarkson Potter 2011. Provided courtesy of Anne Burrell. All rights reserved.