For the pesto: Preheat an oven to 375 degrees F.
Put the peppers on a baking sheet, rub them with the canola oil and season with salt and black pepper, to taste. Roast until they are slightly charred and soft, turning several times, about 25 to 30 minutes. Transfer the peppers to a bowl, cover with plastic wrap and let them steam for 10 minutes. Remove the skins and seeds.
Put the peppers, garlic, basil and pine nuts into the bowl of a food processor and pulse until coarsely chopped. With the motor running, slowly pour the olive oil through the feed tube and pulse to emulsify. Add the cheese and salt and pepper, to taste and pulse a few more times just to combine. Scrape the mixture into a bowl.
For the pizza: Light a grill to high heat.
Roll the pizza dough into 2 (12-inch) circles, brush the top with some canola oil, and season them with salt and black pepper, to taste. Put them on the grill, oiled side down, and grill until the bottoms are slightly charred and golden brown. Brush the tops with additional oil before flipping the pizzas over. Continue grilling until the crusts are just firm, about 40 seconds.
Transfer the shells to a baking sheet, top each with some of the pesto, mozzarella and shaved Parmigiano and return them to the grill. Close the cover and grill until the cheese melts. Remove the pizzas from the grill and finish the pizza with some sliced red onion and handful of arugula. Slice and serve.
Combine the bread flour, sugar, yeast and kosher salt in the bowl of a stand mixer and combine. While the mixer is running, add the water and 2 tablespoons of the oil and beat until the dough forms into a ball. If the dough is sticky, add additional flour, 1 tablespoon at a time, until the dough comes together in a solid ball. If the dough is too dry, add additional water, 1 tablespoon at a time. Scrape the dough onto a lightly floured surface and gently knead into a smooth, firm ball.
Grease a large bowl with the remaining 2 teaspoons olive oil, add the dough, cover the bowl with plastic wrap and put it in a warm area to let it double in size, about 1 hour. Turn the dough out onto a lightly floured surface and divide it into 2 equal pieces. Cover each with a clean kitchen towel or plastic wrap and let them rest for 10 minutes.